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❍✉r❡✇✐❝③ s♣❛❝❡s ✐♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧

▲②✉❜♦♠②r ❩❞♦♠s❦②②

❑✉rt ●ö❞❡❧ ❘❡s❡❛r❝❤ ❈❡♥t❡r ❢♦r ▼❛t❤❡♠❛t✐❝❛❧ ▲♦❣✐❝ ❯♥✐✈❡rs✐t② ♦❢ ❱✐❡♥♥❛

◆♦✈✐ ❙❛❞✱ ❏✉♥❡ ✷✶✱ ✷✵✶✻

✶ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ r❡❧❛t✐✈❡s

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ ❝♦✈❡r ♦❢ X✳ ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ✐s ❍✉r❡✇✐❝③ ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t ❛♥❞ ✐s ❛ ✲❝♦✈❡r ♦❢ ✳ ✐s ❛ ✲❝♦✈❡r ♦❢ ✐❢ ✳ ✲❝♦♠♣❛❝t ❍✉r❡✇✐❝③ ▼❡♥❣❡r ▲✐♥❞❡❧ö❢✳ ❊①❛♠♣❧❡✿ ✐s ♥♦t ▼❡♥❣❡r✳ ❲✐t♥❡ss✿ ✳ ❋♦❧❦❧♦r❡ ❋❛❝t✳ ❋♦r ❛♥❛❧②t✐❝ s❡ts ♦❢ r❡❛❧s ▼❡♥❣❡r ✐s ❡q✉✐✈❛❧❡♥t t♦ ✲❝♦♠♣❛❝t✳ ■♥ t❤❡r❡ ❡①✐sts ❛ ❝♦✲❛♥❛❧②t✐❝ ▼❡♥❣❡r s✉❜s♣❛❝❡ ♦❢ ✇❤✐❝❤ ✐s ♥♦t ✲❝♦♠♣❛❝t✳

✷ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ r❡❧❛t✐✈❡s

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ ❝♦✈❡r ♦❢ X✳ ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ γ✲❝♦✈❡r ♦❢ X✳ ✐s ❛ ✲❝♦✈❡r ♦❢ ✐❢ ✳ ✲❝♦♠♣❛❝t ❍✉r❡✇✐❝③ ▼❡♥❣❡r ▲✐♥❞❡❧ö❢✳ ❊①❛♠♣❧❡✿ ✐s ♥♦t ▼❡♥❣❡r✳ ❲✐t♥❡ss✿ ✳ ❋♦❧❦❧♦r❡ ❋❛❝t✳ ❋♦r ❛♥❛❧②t✐❝ s❡ts ♦❢ r❡❛❧s ▼❡♥❣❡r ✐s ❡q✉✐✈❛❧❡♥t t♦ ✲❝♦♠♣❛❝t✳ ■♥ t❤❡r❡ ❡①✐sts ❛ ❝♦✲❛♥❛❧②t✐❝ ▼❡♥❣❡r s✉❜s♣❛❝❡ ♦❢ ✇❤✐❝❤ ✐s ♥♦t ✲❝♦♠♣❛❝t✳

✷ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ r❡❧❛t✐✈❡s

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ ❝♦✈❡r ♦❢ X✳ ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ γ✲❝♦✈❡r ♦❢ X✳ U ✐s ❛ γ✲❝♦✈❡r ♦❢ X ✐❢ ∀x ∈ X∀∗U ∈ U(x ∈ U)✳ ✲❝♦♠♣❛❝t ❍✉r❡✇✐❝③ ▼❡♥❣❡r ▲✐♥❞❡❧ö❢✳ ❊①❛♠♣❧❡✿ ✐s ♥♦t ▼❡♥❣❡r✳ ❲✐t♥❡ss✿ ✳ ❋♦❧❦❧♦r❡ ❋❛❝t✳ ❋♦r ❛♥❛❧②t✐❝ s❡ts ♦❢ r❡❛❧s ▼❡♥❣❡r ✐s ❡q✉✐✈❛❧❡♥t t♦ ✲❝♦♠♣❛❝t✳ ■♥ t❤❡r❡ ❡①✐sts ❛ ❝♦✲❛♥❛❧②t✐❝ ▼❡♥❣❡r s✉❜s♣❛❝❡ ♦❢ ✇❤✐❝❤ ✐s ♥♦t ✲❝♦♠♣❛❝t✳

✷ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ r❡❧❛t✐✈❡s

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ ❝♦✈❡r ♦❢ X✳ ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ γ✲❝♦✈❡r ♦❢ X✳ U ✐s ❛ γ✲❝♦✈❡r ♦❢ X ✐❢ ∀x ∈ X∀∗U ∈ U(x ∈ U)✳ σ✲❝♦♠♣❛❝t → ❍✉r❡✇✐❝③ → ▼❡♥❣❡r → ▲✐♥❞❡❧ö❢✳ ❊①❛♠♣❧❡✿ ✐s ♥♦t ▼❡♥❣❡r✳ ❲✐t♥❡ss✿ ✳ ❋♦❧❦❧♦r❡ ❋❛❝t✳ ❋♦r ❛♥❛❧②t✐❝ s❡ts ♦❢ r❡❛❧s ▼❡♥❣❡r ✐s ❡q✉✐✈❛❧❡♥t t♦ ✲❝♦♠♣❛❝t✳ ■♥ t❤❡r❡ ❡①✐sts ❛ ❝♦✲❛♥❛❧②t✐❝ ▼❡♥❣❡r s✉❜s♣❛❝❡ ♦❢ ✇❤✐❝❤ ✐s ♥♦t ✲❝♦♠♣❛❝t✳

✷ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ r❡❧❛t✐✈❡s

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ ❝♦✈❡r ♦❢ X✳ ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ γ✲❝♦✈❡r ♦❢ X✳ U ✐s ❛ γ✲❝♦✈❡r ♦❢ X ✐❢ ∀x ∈ X∀∗U ∈ U(x ∈ U)✳ σ✲❝♦♠♣❛❝t → ❍✉r❡✇✐❝③ → ▼❡♥❣❡r → ▲✐♥❞❡❧ö❢✳ ❊①❛♠♣❧❡✿ ωω ✐s ♥♦t ▼❡♥❣❡r✳ ❲✐t♥❡ss✿ ✳ ❋♦❧❦❧♦r❡ ❋❛❝t✳ ❋♦r ❛♥❛❧②t✐❝ s❡ts ♦❢ r❡❛❧s ▼❡♥❣❡r ✐s ❡q✉✐✈❛❧❡♥t t♦ ✲❝♦♠♣❛❝t✳ ■♥ t❤❡r❡ ❡①✐sts ❛ ❝♦✲❛♥❛❧②t✐❝ ▼❡♥❣❡r s✉❜s♣❛❝❡ ♦❢ ✇❤✐❝❤ ✐s ♥♦t ✲❝♦♠♣❛❝t✳

✷ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ r❡❧❛t✐✈❡s

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ ❝♦✈❡r ♦❢ X✳ ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ γ✲❝♦✈❡r ♦❢ X✳ U ✐s ❛ γ✲❝♦✈❡r ♦❢ X ✐❢ ∀x ∈ X∀∗U ∈ U(x ∈ U)✳ σ✲❝♦♠♣❛❝t → ❍✉r❡✇✐❝③ → ▼❡♥❣❡r → ▲✐♥❞❡❧ö❢✳ ❊①❛♠♣❧❡✿ ωω ✐s ♥♦t ▼❡♥❣❡r✳ ❲✐t♥❡ss✿ Un =

• {x : x(n) = k} : k ∈ ω
• ✳

❋♦❧❦❧♦r❡ ❋❛❝t✳ ❋♦r ❛♥❛❧②t✐❝ s❡ts ♦❢ r❡❛❧s ▼❡♥❣❡r ✐s ❡q✉✐✈❛❧❡♥t t♦ ✲❝♦♠♣❛❝t✳ ■♥ t❤❡r❡ ❡①✐sts ❛ ❝♦✲❛♥❛❧②t✐❝ ▼❡♥❣❡r s✉❜s♣❛❝❡ ♦❢ ✇❤✐❝❤ ✐s ♥♦t ✲❝♦♠♣❛❝t✳

✷ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ r❡❧❛t✐✈❡s

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ ❝♦✈❡r ♦❢ X✳ ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ γ✲❝♦✈❡r ♦❢ X✳ U ✐s ❛ γ✲❝♦✈❡r ♦❢ X ✐❢ ∀x ∈ X∀∗U ∈ U(x ∈ U)✳ σ✲❝♦♠♣❛❝t → ❍✉r❡✇✐❝③ → ▼❡♥❣❡r → ▲✐♥❞❡❧ö❢✳ ❊①❛♠♣❧❡✿ ωω ✐s ♥♦t ▼❡♥❣❡r✳ ❲✐t♥❡ss✿ Un =

• {x : x(n) = k} : k ∈ ω
• ✳

❋♦❧❦❧♦r❡ ❋❛❝t✳ ❋♦r ❛♥❛❧②t✐❝ s❡ts ♦❢ r❡❛❧s ▼❡♥❣❡r ✐s ❡q✉✐✈❛❧❡♥t t♦ σ✲❝♦♠♣❛❝t✳ ■♥ t❤❡r❡ ❡①✐sts ❛ ❝♦✲❛♥❛❧②t✐❝ ▼❡♥❣❡r s✉❜s♣❛❝❡ ♦❢ ✇❤✐❝❤ ✐s ♥♦t ✲❝♦♠♣❛❝t✳

✷ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ r❡❧❛t✐✈❡s

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ ❝♦✈❡r ♦❢ X✳ ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Un : n ∈ ω ♦❢ ♦♣❡♥ ❝♦✈❡rs ♦❢ X t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ Vn : n ∈ ω s✉❝❤ t❤❛t Vn ∈ [Un]<ω ❛♥❞ {∪Vn : n ∈ ω} ✐s ❛ γ✲❝♦✈❡r ♦❢ X✳ U ✐s ❛ γ✲❝♦✈❡r ♦❢ X ✐❢ ∀x ∈ X∀∗U ∈ U(x ∈ U)✳ σ✲❝♦♠♣❛❝t → ❍✉r❡✇✐❝③ → ▼❡♥❣❡r → ▲✐♥❞❡❧ö❢✳ ❊①❛♠♣❧❡✿ ωω ✐s ♥♦t ▼❡♥❣❡r✳ ❲✐t♥❡ss✿ Un =

• {x : x(n) = k} : k ∈ ω
• ✳

❋♦❧❦❧♦r❡ ❋❛❝t✳ ❋♦r ❛♥❛❧②t✐❝ s❡ts ♦❢ r❡❛❧s ▼❡♥❣❡r ✐s ❡q✉✐✈❛❧❡♥t t♦ σ✲❝♦♠♣❛❝t✳ ■♥ L t❤❡r❡ ❡①✐sts ❛ ❝♦✲❛♥❛❧②t✐❝ ▼❡♥❣❡r s✉❜s♣❛❝❡ ♦❢ ωω ✇❤✐❝❤ ✐s ♥♦t σ✲❝♦♠♣❛❝t✳

✷ ✴ ✶✽

❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥

• ✐✈❡♥ x, y ∈ ωω✱ x ≤∗ y ♠❡❛♥s {n : x(n) ≤ y(n)} ✐s ❝♦✜♥✐t❡✳

❚❤❡♦r❡♠ ✭❍✉r❡✇✐❝③ ✶✾✷✺✮

❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ ✐s ❍✉r❡✇✐❝③ ✐✛ ✐s ❜♦✉♥❞❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s ✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ ✐s ▼❡♥❣❡r ✐✛ ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s ✳ ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ✳ ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ s✉❜s❡t ♦❢ ✳ ✐s ❍✉r❡✇✐❝③✳ ✐s ▼❡♥❣❡r✳

✸ ✴ ✶✽

❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥

• ✐✈❡♥ x, y ∈ ωω✱ x ≤∗ y ♠❡❛♥s {n : x(n) ≤ y(n)} ✐s ❝♦✜♥✐t❡✳

❚❤❡♦r❡♠ ✭❍✉r❡✇✐❝③ ✶✾✷✺✮

❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐✛ f[X] ✐s ❜♦✉♥❞❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ≤∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ ✐s ▼❡♥❣❡r ✐✛ ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s ✳ ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ✳ ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ s✉❜s❡t ♦❢ ✳ ✐s ❍✉r❡✇✐❝③✳ ✐s ▼❡♥❣❡r✳

✸ ✴ ✶✽

❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥

• ✐✈❡♥ x, y ∈ ωω✱ x ≤∗ y ♠❡❛♥s {n : x(n) ≤ y(n)} ✐s ❝♦✜♥✐t❡✳

❚❤❡♦r❡♠ ✭❍✉r❡✇✐❝③ ✶✾✷✺✮

❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐✛ f[X] ✐s ❜♦✉♥❞❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ≤∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f[X] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ≤∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ✳ ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ s✉❜s❡t ♦❢ ✳ ✐s ❍✉r❡✇✐❝③✳ ✐s ▼❡♥❣❡r✳

✸ ✴ ✶✽

❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥

• ✐✈❡♥ x, y ∈ ωω✱ x ≤∗ y ♠❡❛♥s {n : x(n) ≤ y(n)} ✐s ❝♦✜♥✐t❡✳

❚❤❡♦r❡♠ ✭❍✉r❡✇✐❝③ ✶✾✷✺✮

❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐✛ f[X] ✐s ❜♦✉♥❞❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ≤∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f[X] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ≤∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ b ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ωω✳ ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ s✉❜s❡t ♦❢ ✳ ✐s ❍✉r❡✇✐❝③✳ ✐s ▼❡♥❣❡r✳

✸ ✴ ✶✽

❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥

• ✐✈❡♥ x, y ∈ ωω✱ x ≤∗ y ♠❡❛♥s {n : x(n) ≤ y(n)} ✐s ❝♦✜♥✐t❡✳

❚❤❡♦r❡♠ ✭❍✉r❡✇✐❝③ ✶✾✷✺✮

❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐✛ f[X] ✐s ❜♦✉♥❞❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ≤∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f[X] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ≤∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ b ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ωω✳ d ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ s✉❜s❡t ♦❢ ωω✳ ✐s ❍✉r❡✇✐❝③✳ ✐s ▼❡♥❣❡r✳

✸ ✴ ✶✽

❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥

• ✐✈❡♥ x, y ∈ ωω✱ x ≤∗ y ♠❡❛♥s {n : x(n) ≤ y(n)} ✐s ❝♦✜♥✐t❡✳

❚❤❡♦r❡♠ ✭❍✉r❡✇✐❝③ ✶✾✷✺✮

❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐✛ f[X] ✐s ❜♦✉♥❞❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ≤∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f[X] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ≤∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ b ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ωω✳ d ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ s✉❜s❡t ♦❢ ωω✳ |X| < b → X ✐s ❍✉r❡✇✐❝③✳ ✐s ▼❡♥❣❡r✳

✸ ✴ ✶✽

❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥

• ✐✈❡♥ x, y ∈ ωω✱ x ≤∗ y ♠❡❛♥s {n : x(n) ≤ y(n)} ✐s ❝♦✜♥✐t❡✳

❚❤❡♦r❡♠ ✭❍✉r❡✇✐❝③ ✶✾✷✺✮

❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐✛ f[X] ✐s ❜♦✉♥❞❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ≤∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f[X] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ≤∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ b ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ωω✳ d ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ s✉❜s❡t ♦❢ ωω✳ |X| < b → X ✐s ❍✉r❡✇✐❝③✳ |X| < d → X ✐s ▼❡♥❣❡r✳

✸ ✴ ✶✽

❊①❛♠♣❧❡s ✉♥❞❡r ❈❍✳

X ⊂ ωω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ M| ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ ❛♥❞ ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t ✳

❚❤❡♦r❡♠ ✭❙❝❤❡❡♣❡rs ✶✾✾✻✮

▲❡t ❜❡ ❝♦♠♣❛❝t✳ ✐s ❍✉r❡✇✐❝③ ✐✛ ❢♦r ❡✈❡r② ✲s❡t t❤❡r❡ ❡①✐sts ❛ ✲❝♦♠♣❛❝t s✉❝❤ t❤❛t ✳

❈♦r♦❧❧❛r②

▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r ❜✉t ♥♦t ❍✉r❡✇✐❝③✳ ❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✿ ✲❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③ ❛♥❞ ✲▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r✳

✹ ✴ ✶✽

❊①❛♠♣❧❡s ✉♥❞❡r ❈❍✳

X ⊂ ωω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ M| ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ ❛♥❞ ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t ✳

❚❤❡♦r❡♠ ✭❙❝❤❡❡♣❡rs ✶✾✾✻✮

▲❡t ❜❡ ❝♦♠♣❛❝t✳ ✐s ❍✉r❡✇✐❝③ ✐✛ ❢♦r ❡✈❡r② ✲s❡t t❤❡r❡ ❡①✐sts ❛ ✲❝♦♠♣❛❝t s✉❝❤ t❤❛t ✳

❈♦r♦❧❧❛r②

▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r ❜✉t ♥♦t ❍✉r❡✇✐❝③✳ ❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✿ ✲❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③ ❛♥❞ ✲▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r✳

✹ ✴ ✶✽

❊①❛♠♣❧❡s ✉♥❞❡r ❈❍✳

X ⊂ ωω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ M| ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ X ⊂ 2ω ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ N| ≤ ω ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t N✳

❚❤❡♦r❡♠ ✭❙❝❤❡❡♣❡rs ✶✾✾✻✮

▲❡t ❜❡ ❝♦♠♣❛❝t✳ ✐s ❍✉r❡✇✐❝③ ✐✛ ❢♦r ❡✈❡r② ✲s❡t t❤❡r❡ ❡①✐sts ❛ ✲❝♦♠♣❛❝t s✉❝❤ t❤❛t ✳

❈♦r♦❧❧❛r②

▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r ❜✉t ♥♦t ❍✉r❡✇✐❝③✳ ❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✿ ✲❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③ ❛♥❞ ✲▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r✳

✹ ✴ ✶✽

❊①❛♠♣❧❡s ✉♥❞❡r ❈❍✳

X ⊂ ωω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ M| ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ X ⊂ 2ω ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ N| ≤ ω ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t N✳

❚❤❡♦r❡♠ ✭❙❝❤❡❡♣❡rs ✶✾✾✻✮

▲❡t P ❜❡ ❝♦♠♣❛❝t✳ X ⊂ P ✐s ❍✉r❡✇✐❝③ ✐✛ ❢♦r ❡✈❡r② Gδ✲s❡t G ⊃ X t❤❡r❡ ❡①✐sts ❛ σ✲❝♦♠♣❛❝t F s✉❝❤ t❤❛t X ⊂ F ⊂ G✳

❈♦r♦❧❧❛r②

▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r ❜✉t ♥♦t ❍✉r❡✇✐❝③✳ ❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✿ ✲❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③ ❛♥❞ ✲▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r✳

✹ ✴ ✶✽

❊①❛♠♣❧❡s ✉♥❞❡r ❈❍✳

X ⊂ ωω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ M| ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ X ⊂ 2ω ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ N| ≤ ω ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t N✳

❚❤❡♦r❡♠ ✭❙❝❤❡❡♣❡rs ✶✾✾✻✮

▲❡t P ❜❡ ❝♦♠♣❛❝t✳ X ⊂ P ✐s ❍✉r❡✇✐❝③ ✐✛ ❢♦r ❡✈❡r② Gδ✲s❡t G ⊃ X t❤❡r❡ ❡①✐sts ❛ σ✲❝♦♠♣❛❝t F s✉❝❤ t❤❛t X ⊂ F ⊂ G✳

❈♦r♦❧❧❛r②

▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r ❜✉t ♥♦t ❍✉r❡✇✐❝③✳ ❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✿ ✲❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③ ❛♥❞ ✲▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r✳

✹ ✴ ✶✽

❊①❛♠♣❧❡s ✉♥❞❡r ❈❍✳

X ⊂ ωω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ M| ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ X ⊂ 2ω ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ N| ≤ ω ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t N✳

❚❤❡♦r❡♠ ✭❙❝❤❡❡♣❡rs ✶✾✾✻✮

▲❡t P ❜❡ ❝♦♠♣❛❝t✳ X ⊂ P ✐s ❍✉r❡✇✐❝③ ✐✛ ❢♦r ❡✈❡r② Gδ✲s❡t G ⊃ X t❤❡r❡ ❡①✐sts ❛ σ✲❝♦♠♣❛❝t F s✉❝❤ t❤❛t X ⊂ F ⊂ G✳

❈♦r♦❧❧❛r②

▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r ❜✉t ♥♦t ❍✉r❡✇✐❝③✳ ❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✿ ✲❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③ ❛♥❞ ✲▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r✳

✹ ✴ ✶✽

❊①❛♠♣❧❡s ✉♥❞❡r ❈❍✳

X ⊂ ωω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ M| ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ X ⊂ 2ω ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ |X| = ω1 ❛♥❞ |X ∩ N| ≤ ω ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t N✳

❚❤❡♦r❡♠ ✭❙❝❤❡❡♣❡rs ✶✾✾✻✮

▲❡t P ❜❡ ❝♦♠♣❛❝t✳ X ⊂ P ✐s ❍✉r❡✇✐❝③ ✐✛ ❢♦r ❡✈❡r② Gδ✲s❡t G ⊃ X t❤❡r❡ ❡①✐sts ❛ σ✲❝♦♠♣❛❝t F s✉❝❤ t❤❛t X ⊂ F ⊂ G✳

❈♦r♦❧❧❛r②

▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r ❜✉t ♥♦t ❍✉r❡✇✐❝③✳ ❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✿ b✲❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③ ❛♥❞ d✲▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r✳

✹ ✴ ✶✽

❩❋❈ ❡①❛♠♣❧❡s

❆ s❡t X ⊂ ωω ✐s κ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q✱ ✐❢ |X| ≥ κ ❛♥❞ |X \ U| < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ωω ❝♦♥t❛✐♥✐♥❣ Q✳ ■❢ ✱ t❤❡♥ ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✳ ❱✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ ✱ ✐s ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ✐s ❡✈❡♥t✉❛❧❧② ✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✳ ✐s ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ✳ ◆♦♥tr✐✈✐❛❧ ✭❇❛rt♦s③②♥s❦✐✲❙❤❡❧❛❤✮✿ ✐s ❍✉r❡✇✐❝③✳ ✑❆❧❧ ✲❝♦♥❝❡♥tr❛t❡❞ s❡ts ❛r❡ ❍✉r❡✇✐❝③✑ ✐s ✐♥❞❡♣❡♥❞❡♥t✿ ✇r♦♥❣ ✉♥❞❡r ❈❍✱ tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳

✺ ✴ ✶✽

❩❋❈ ❡①❛♠♣❧❡s

❆ s❡t X ⊂ ωω ✐s κ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q✱ ✐❢ |X| ≥ κ ❛♥❞ |X \ U| < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ωω ❝♦♥t❛✐♥✐♥❣ Q✳ ■❢ κ ≤ d✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✳ ❱✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ ✱ ✐s ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ✐s ❡✈❡♥t✉❛❧❧② ✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✳ ✐s ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ✳ ◆♦♥tr✐✈✐❛❧ ✭❇❛rt♦s③②♥s❦✐✲❙❤❡❧❛❤✮✿ ✐s ❍✉r❡✇✐❝③✳ ✑❆❧❧ ✲❝♦♥❝❡♥tr❛t❡❞ s❡ts ❛r❡ ❍✉r❡✇✐❝③✑ ✐s ✐♥❞❡♣❡♥❞❡♥t✿ ✇r♦♥❣ ✉♥❞❡r ❈❍✱ tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳

✺ ✴ ✶✽

❩❋❈ ❡①❛♠♣❧❡s

❆ s❡t X ⊂ ωω ✐s κ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q✱ ✐❢ |X| ≥ κ ❛♥❞ |X \ U| < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ωω ❝♦♥t❛✐♥✐♥❣ Q✳ ■❢ κ ≤ d✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ {dα : α < d} ⊂ ωω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = {sα : α < d} ⊂ ω↑ω s✉❝❤ t❤❛t sα ≤∗ dβ ❢♦r ❛❧❧ β ≤ α✳ ❱✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ ✱ ✐s ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ✐s ❡✈❡♥t✉❛❧❧② ✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✳ ✐s ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ✳ ◆♦♥tr✐✈✐❛❧ ✭❇❛rt♦s③②♥s❦✐✲❙❤❡❧❛❤✮✿ ✐s ❍✉r❡✇✐❝③✳ ✑❆❧❧ ✲❝♦♥❝❡♥tr❛t❡❞ s❡ts ❛r❡ ❍✉r❡✇✐❝③✑ ✐s ✐♥❞❡♣❡♥❞❡♥t✿ ✇r♦♥❣ ✉♥❞❡r ❈❍✱ tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳

✺ ✴ ✶✽

❩❋❈ ❡①❛♠♣❧❡s

❆ s❡t X ⊂ ωω ✐s κ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q✱ ✐❢ |X| ≥ κ ❛♥❞ |X \ U| < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ωω ❝♦♥t❛✐♥✐♥❣ Q✳ ■❢ κ ≤ d✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ {dα : α < d} ⊂ ωω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = {sα : α < d} ⊂ ω↑ω s✉❝❤ t❤❛t sα ≤∗ dβ ❢♦r ❛❧❧ β ≤ α✳ ❱✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ (ω + 1)↑ω✱ S ✐s d✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q = {x ∈ (ω + 1)↑ω : x ✐s ❡✈❡♥t✉❛❧❧② ω}✳ ✷ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✳ ✐s ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ✳ ◆♦♥tr✐✈✐❛❧ ✭❇❛rt♦s③②♥s❦✐✲❙❤❡❧❛❤✮✿ ✐s ❍✉r❡✇✐❝③✳ ✑❆❧❧ ✲❝♦♥❝❡♥tr❛t❡❞ s❡ts ❛r❡ ❍✉r❡✇✐❝③✑ ✐s ✐♥❞❡♣❡♥❞❡♥t✿ ✇r♦♥❣ ✉♥❞❡r ❈❍✱ tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳

✺ ✴ ✶✽

❩❋❈ ❡①❛♠♣❧❡s

❆ s❡t X ⊂ ωω ✐s κ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q✱ ✐❢ |X| ≥ κ ❛♥❞ |X \ U| < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ωω ❝♦♥t❛✐♥✐♥❣ Q✳ ■❢ κ ≤ d✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ {dα : α < d} ⊂ ωω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = {sα : α < d} ⊂ ω↑ω s✉❝❤ t❤❛t sα ≤∗ dβ ❢♦r ❛❧❧ β ≤ α✳ ❱✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ (ω + 1)↑ω✱ S ✐s d✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q = {x ∈ (ω + 1)↑ω : x ✐s ❡✈❡♥t✉❛❧❧② ω}✳ ✷ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ b✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛♥ ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bβ ≤∗ bα ❢♦r ❛❧❧ β ≤ α✳ ✐s ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ✳ ◆♦♥tr✐✈✐❛❧ ✭❇❛rt♦s③②♥s❦✐✲❙❤❡❧❛❤✮✿ ✐s ❍✉r❡✇✐❝③✳ ✑❆❧❧ ✲❝♦♥❝❡♥tr❛t❡❞ s❡ts ❛r❡ ❍✉r❡✇✐❝③✑ ✐s ✐♥❞❡♣❡♥❞❡♥t✿ ✇r♦♥❣ ✉♥❞❡r ❈❍✱ tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳

✺ ✴ ✶✽

❩❋❈ ❡①❛♠♣❧❡s

❆ s❡t X ⊂ ωω ✐s κ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q✱ ✐❢ |X| ≥ κ ❛♥❞ |X \ U| < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ωω ❝♦♥t❛✐♥✐♥❣ Q✳ ■❢ κ ≤ d✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ {dα : α < d} ⊂ ωω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = {sα : α < d} ⊂ ω↑ω s✉❝❤ t❤❛t sα ≤∗ dβ ❢♦r ❛❧❧ β ≤ α✳ ❱✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ (ω + 1)↑ω✱ S ✐s d✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q = {x ∈ (ω + 1)↑ω : x ✐s ❡✈❡♥t✉❛❧❧② ω}✳ ✷ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ b✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛♥ ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bβ ≤∗ bα ❢♦r ❛❧❧ β ≤ α✳ B ✐s b✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q✳ ✷ ◆♦♥tr✐✈✐❛❧ ✭❇❛rt♦s③②♥s❦✐✲❙❤❡❧❛❤✮✿ ✐s ❍✉r❡✇✐❝③✳ ✑❆❧❧ ✲❝♦♥❝❡♥tr❛t❡❞ s❡ts ❛r❡ ❍✉r❡✇✐❝③✑ ✐s ✐♥❞❡♣❡♥❞❡♥t✿ ✇r♦♥❣ ✉♥❞❡r ❈❍✱ tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳

✺ ✴ ✶✽

❩❋❈ ❡①❛♠♣❧❡s

❆ s❡t X ⊂ ωω ✐s κ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q✱ ✐❢ |X| ≥ κ ❛♥❞ |X \ U| < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ωω ❝♦♥t❛✐♥✐♥❣ Q✳ ■❢ κ ≤ d✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ {dα : α < d} ⊂ ωω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = {sα : α < d} ⊂ ω↑ω s✉❝❤ t❤❛t sα ≤∗ dβ ❢♦r ❛❧❧ β ≤ α✳ ❱✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ (ω + 1)↑ω✱ S ✐s d✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q = {x ∈ (ω + 1)↑ω : x ✐s ❡✈❡♥t✉❛❧❧② ω}✳ ✷ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ b✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛♥ ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bβ ≤∗ bα ❢♦r ❛❧❧ β ≤ α✳ B ✐s b✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q✳ ✷ ◆♦♥tr✐✈✐❛❧ ✭❇❛rt♦s③②♥s❦✐✲❙❤❡❧❛❤✮✿ B ∪ Q ✐s ❍✉r❡✇✐❝③✳ ✑❆❧❧ ✲❝♦♥❝❡♥tr❛t❡❞ s❡ts ❛r❡ ❍✉r❡✇✐❝③✑ ✐s ✐♥❞❡♣❡♥❞❡♥t✿ ✇r♦♥❣ ✉♥❞❡r ❈❍✱ tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳

✺ ✴ ✶✽

❩❋❈ ❡①❛♠♣❧❡s

❆ s❡t X ⊂ ωω ✐s κ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q✱ ✐❢ |X| ≥ κ ❛♥❞ |X \ U| < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ωω ❝♦♥t❛✐♥✐♥❣ Q✳ ■❢ κ ≤ d✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ {dα : α < d} ⊂ ωω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = {sα : α < d} ⊂ ω↑ω s✉❝❤ t❤❛t sα ≤∗ dβ ❢♦r ❛❧❧ β ≤ α✳ ❱✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ (ω + 1)↑ω✱ S ✐s d✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q = {x ∈ (ω + 1)↑ω : x ✐s ❡✈❡♥t✉❛❧❧② ω}✳ ✷ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ b✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛♥ ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bβ ≤∗ bα ❢♦r ❛❧❧ β ≤ α✳ B ✐s b✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q✳ ✷ ◆♦♥tr✐✈✐❛❧ ✭❇❛rt♦s③②♥s❦✐✲❙❤❡❧❛❤✮✿ B ∪ Q ✐s ❍✉r❡✇✐❝③✳ ✑❆❧❧ b✲❝♦♥❝❡♥tr❛t❡❞ s❡ts ❛r❡ ❍✉r❡✇✐❝③✑ ✐s ✐♥❞❡♣❡♥❞❡♥t✿ ✇r♦♥❣ ✉♥❞❡r ❈❍✱ tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳

✺ ✴ ✶✽

❩❋❈ ❡①❛♠♣❧❡s

❆ s❡t X ⊂ ωω ✐s κ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q✱ ✐❢ |X| ≥ κ ❛♥❞ |X \ U| < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ωω ❝♦♥t❛✐♥✐♥❣ Q✳ ■❢ κ ≤ d✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ {dα : α < d} ⊂ ωω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = {sα : α < d} ⊂ ω↑ω s✉❝❤ t❤❛t sα ≤∗ dβ ❢♦r ❛❧❧ β ≤ α✳ ❱✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ (ω + 1)↑ω✱ S ✐s d✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q = {x ∈ (ω + 1)↑ω : x ✐s ❡✈❡♥t✉❛❧❧② ω}✳ ✷ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ b✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛♥ ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bβ ≤∗ bα ❢♦r ❛❧❧ β ≤ α✳ B ✐s b✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q✳ ✷ ◆♦♥tr✐✈✐❛❧ ✭❇❛rt♦s③②♥s❦✐✲❙❤❡❧❛❤✮✿ B ∪ Q ✐s ❍✉r❡✇✐❝③✳ ✑❆❧❧ b✲❝♦♥❝❡♥tr❛t❡❞ s❡ts ❛r❡ ❍✉r❡✇✐❝③✑ ✐s ✐♥❞❡♣❡♥❞❡♥t✿ ✇r♦♥❣ ✉♥❞❡r ❈❍✱ tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳

✺ ✴ ✶✽

❩❋❈ ❡①❛♠♣❧❡s

❆ s❡t X ⊂ ωω ✐s κ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q✱ ✐❢ |X| ≥ κ ❛♥❞ |X \ U| < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ωω ❝♦♥t❛✐♥✐♥❣ Q✳ ■❢ κ ≤ d✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ {dα : α < d} ⊂ ωω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = {sα : α < d} ⊂ ω↑ω s✉❝❤ t❤❛t sα ≤∗ dβ ❢♦r ❛❧❧ β ≤ α✳ ❱✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ (ω + 1)↑ω✱ S ✐s d✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q = {x ∈ (ω + 1)↑ω : x ✐s ❡✈❡♥t✉❛❧❧② ω}✳ ✷ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ b✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛♥ ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bβ ≤∗ bα ❢♦r ❛❧❧ β ≤ α✳ B ✐s b✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q✳ ✷ ◆♦♥tr✐✈✐❛❧ ✭❇❛rt♦s③②♥s❦✐✲❙❤❡❧❛❤✮✿ B ∪ Q ✐s ❍✉r❡✇✐❝③✳ ✑❆❧❧ b✲❝♦♥❝❡♥tr❛t❡❞ s❡ts ❛r❡ ❍✉r❡✇✐❝③✑ ✐s ✐♥❞❡♣❡♥❞❡♥t✿ ✇r♦♥❣ ✉♥❞❡r ❈❍✱ tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳

✺ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ ❝♦♥s✐sts ♦❢ ♣❛✐rs s✉❝❤ t❤❛t ✱ ✱ ❛♥❞ ✳ ❆ ❝♦♥❞✐t✐♦♥ ✐s str♦♥❣❡r t❤❛♥ ✐❢ ✱ ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ ✱ ❛♥❞ ✳ ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✳ ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ ✿ ✐❢ ✐s ❛ ✲❣❡♥❡r✐❝✱ t❤❡♥ ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ ❝♦♥s✐sts ♦❢ ♣❛✐rs s✉❝❤ t❤❛t ✱ ✱ ❛♥❞ ✳ ❆ ❝♦♥❞✐t✐♦♥ ✐s str♦♥❣❡r t❤❛♥ ✐❢ ✱ ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ ✱ ❛♥❞ ✳ ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✳ ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ ✿ ✐❢ ✐s ❛ ✲❣❡♥❡r✐❝✱ t❤❡♥ ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ ❝♦♥s✐sts ♦❢ ♣❛✐rs s✉❝❤ t❤❛t ✱ ✱ ❛♥❞ ✳ ❆ ❝♦♥❞✐t✐♦♥ ✐s str♦♥❣❡r t❤❛♥ ✐❢ ✱ ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ ✱ ❛♥❞ ✳ ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✳ ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ ✿ ✐❢ ✐s ❛ ✲❣❡♥❡r✐❝✱ t❤❡♥ ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ MF ❝♦♥s✐sts ♦❢ ♣❛✐rs s, F s✉❝❤ t❤❛t s ∈ [ω]<ω✱ F ∈ F✱ ❛♥❞ max s < min F✳ ❆ ❝♦♥❞✐t✐♦♥ ✐s str♦♥❣❡r t❤❛♥ ✐❢ ✱ ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ ✱ ❛♥❞ ✳ ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✳ ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ ✿ ✐❢ ✐s ❛ ✲❣❡♥❡r✐❝✱ t❤❡♥ ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ MF ❝♦♥s✐sts ♦❢ ♣❛✐rs s, F s✉❝❤ t❤❛t s ∈ [ω]<ω✱ F ∈ F✱ ❛♥❞ max s < min F✳ ❆ ❝♦♥❞✐t✐♦♥ s, F ✐s str♦♥❣❡r t❤❛♥ t, U ✐❢ F ⊂ U✱ s ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ t✱ ❛♥❞ ✳ ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✳ ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ ✿ ✐❢ ✐s ❛ ✲❣❡♥❡r✐❝✱ t❤❡♥ ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ MF ❝♦♥s✐sts ♦❢ ♣❛✐rs s, F s✉❝❤ t❤❛t s ∈ [ω]<ω✱ F ∈ F✱ ❛♥❞ max s < min F✳ ❆ ❝♦♥❞✐t✐♦♥ s, F ✐s str♦♥❣❡r t❤❛♥ t, U ✐❢ F ⊂ U✱ s ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ t✱ ❛♥❞ s \ t ⊂ U✳ ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✳ ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ ✿ ✐❢ ✐s ❛ ✲❣❡♥❡r✐❝✱ t❤❡♥ ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ MF ❝♦♥s✐sts ♦❢ ♣❛✐rs s, F s✉❝❤ t❤❛t s ∈ [ω]<ω✱ F ∈ F✱ ❛♥❞ max s < min F✳ ❆ ❝♦♥❞✐t✐♦♥ s, F ✐s str♦♥❣❡r t❤❛♥ t, U ✐❢ F ⊂ U✱ s ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ t✱ ❛♥❞ s \ t ⊂ U✳ MF ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F✳ ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ ✿ ✐❢ ✐s ❛ ✲❣❡♥❡r✐❝✱ t❤❡♥ ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ MF ❝♦♥s✐sts ♦❢ ♣❛✐rs s, F s✉❝❤ t❤❛t s ∈ [ω]<ω✱ F ∈ F✱ ❛♥❞ max s < min F✳ ❆ ❝♦♥❞✐t✐♦♥ s, F ✐s str♦♥❣❡r t❤❛♥ t, U ✐❢ F ⊂ U✱ s ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ t✱ ❛♥❞ s \ t ⊂ U✳ MF ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F✳ MF ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ F✿ ✐❢ ✐s ❛ ✲❣❡♥❡r✐❝✱ t❤❡♥ ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ MF ❝♦♥s✐sts ♦❢ ♣❛✐rs s, F s✉❝❤ t❤❛t s ∈ [ω]<ω✱ F ∈ F✱ ❛♥❞ max s < min F✳ ❆ ❝♦♥❞✐t✐♦♥ s, F ✐s str♦♥❣❡r t❤❛♥ t, U ✐❢ F ⊂ U✱ s ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ t✱ ❛♥❞ s \ t ⊂ U✳ MF ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F✳ MF ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ F✿ ✐❢ G ✐s ❛ MF✲❣❡♥❡r✐❝✱ t❤❡♥ X = {s : ∃F ∈ F(s, F ∈ G)} ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② F ∈ F✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ MF ❝♦♥s✐sts ♦❢ ♣❛✐rs s, F s✉❝❤ t❤❛t s ∈ [ω]<ω✱ F ∈ F✱ ❛♥❞ max s < min F✳ ❆ ❝♦♥❞✐t✐♦♥ s, F ✐s str♦♥❣❡r t❤❛♥ t, U ✐❢ F ⊂ U✱ s ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ t✱ ❛♥❞ s \ t ⊂ U✳ MF ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F✳ MF ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ F✿ ✐❢ G ✐s ❛ MF✲❣❡♥❡r✐❝✱ t❤❡♥ X = {s : ∃F ∈ F(s, F ∈ G)} ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② F ∈ F✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ MF ❝♦♥s✐sts ♦❢ ♣❛✐rs s, F s✉❝❤ t❤❛t s ∈ [ω]<ω✱ F ∈ F✱ ❛♥❞ max s < min F✳ ❆ ❝♦♥❞✐t✐♦♥ s, F ✐s str♦♥❣❡r t❤❛♥ t, U ✐❢ F ⊂ U✱ s ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ t✱ ❛♥❞ s \ t ⊂ U✳ MF ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F✳ MF ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ F✿ ✐❢ G ✐s ❛ MF✲❣❡♥❡r✐❝✱ t❤❡♥ X = {s : ∃F ∈ F(s, F ∈ G)} ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② F ∈ F✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

▼❛t❤✐❛s ❢♦r❝✐♥❣ ❢♦r ✜❧t❡rs

❆ s✉❜s❡t F ♦❢ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ✜❧t❡r ✐❢ F ❝♦♥t❛✐♥s ❛❧❧ ❝♦✜♥✐t❡ s❡ts✱ ✐s ❝❧♦s❡❞ ✉♥❞❡r ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ✐ts ❡❧❡♠❡♥ts✱ ❛♥❞ ✉♥❞❡r t❛❦✐♥❣ s✉♣❡rs❡ts✳ MF ❝♦♥s✐sts ♦❢ ♣❛✐rs s, F s✉❝❤ t❤❛t s ∈ [ω]<ω✱ F ∈ F✱ ❛♥❞ max s < min F✳ ❆ ❝♦♥❞✐t✐♦♥ s, F ✐s str♦♥❣❡r t❤❛♥ t, U ✐❢ F ⊂ U✱ s ✐s ❛♥ ❡♥❞✲❡①t❡♥s✐♦♥ ♦❢ t✱ ❛♥❞ s \ t ⊂ U✳ MF ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F✳ MF ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ F✿ ✐❢ G ✐s ❛ MF✲❣❡♥❡r✐❝✱ t❤❡♥ X = {s : ∃F ∈ F(s, F ∈ G)} ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② F ∈ F✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ♠❛❦✐♥❣ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ r❡❛❧s ♥♦t s♣❧✐tt✐♥❣✱ ❡t❝✳

✻ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s

❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ωω s✉❝❤ t❤❛t y ≤∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ωω✳ ✳ ❊①❛♠♣❧❡✿ ▲❛✈❡r ❢♦r❝✐♥❣✱ ❍❡❝❤❧❡r ❢♦r❝✐♥❣✳ ▼✐❧❧❡r ❛♥❞ ❈♦❤❡♥ ❢♦r❝✐♥❣ ❞♦ ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❚❤❡♦r❡♠ ✭❈❛♥❥❛r ✶✾✽✽✮

✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ✉❧tr❛✜❧t❡r s✉❝❤ t❤❛t ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮

❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ✳ ❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ✲❈❛♥❥❛r ✐❢ ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ ✳

✼ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s

❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ωω s✉❝❤ t❤❛t y ≤∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ωω✳ ✳ ❊①❛♠♣❧❡✿ ▲❛✈❡r ❢♦r❝✐♥❣✱ ❍❡❝❤❧❡r ❢♦r❝✐♥❣✳ ▼✐❧❧❡r ❛♥❞ ❈♦❤❡♥ ❢♦r❝✐♥❣ ❞♦ ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❚❤❡♦r❡♠ ✭❈❛♥❥❛r ✶✾✽✽✮

✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ✉❧tr❛✜❧t❡r s✉❝❤ t❤❛t ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮

❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ✳ ❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ✲❈❛♥❥❛r ✐❢ ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ ✳

✼ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s

❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ωω s✉❝❤ t❤❛t y ≤∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ωω✳ ✳ ❊①❛♠♣❧❡✿ ▲❛✈❡r ❢♦r❝✐♥❣✱ ❍❡❝❤❧❡r ❢♦r❝✐♥❣✳ ▼✐❧❧❡r ❛♥❞ ❈♦❤❡♥ ❢♦r❝✐♥❣ ❞♦ ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❚❤❡♦r❡♠ ✭❈❛♥❥❛r ✶✾✽✽✮

✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ✉❧tr❛✜❧t❡r s✉❝❤ t❤❛t ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮

❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ✳ ❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ✲❈❛♥❥❛r ✐❢ ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ ✳

✼ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s

❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ωω s✉❝❤ t❤❛t y ≤∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ωω✳ ✳ ❊①❛♠♣❧❡✿ ▲❛✈❡r ❢♦r❝✐♥❣✱ ❍❡❝❤❧❡r ❢♦r❝✐♥❣✳ ▼✐❧❧❡r ❛♥❞ ❈♦❤❡♥ ❢♦r❝✐♥❣ ❞♦ ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❚❤❡♦r❡♠ ✭❈❛♥❥❛r ✶✾✽✽✮

d = c ✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ✉❧tr❛✜❧t❡r F s✉❝❤ t❤❛t MF ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✷

❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮

❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ✳ ❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ✲❈❛♥❥❛r ✐❢ ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ ✳

✼ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s

❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ωω s✉❝❤ t❤❛t y ≤∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ωω✳ ✳ ❊①❛♠♣❧❡✿ ▲❛✈❡r ❢♦r❝✐♥❣✱ ❍❡❝❤❧❡r ❢♦r❝✐♥❣✳ ▼✐❧❧❡r ❛♥❞ ❈♦❤❡♥ ❢♦r❝✐♥❣ ❞♦ ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❚❤❡♦r❡♠ ✭❈❛♥❥❛r ✶✾✽✽✮

d = c ✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ✉❧tr❛✜❧t❡r F s✉❝❤ t❤❛t MF ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✷

❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮

❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ✳ ❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ✲❈❛♥❥❛r ✐❢ ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ ✳

✼ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s

❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ωω s✉❝❤ t❤❛t y ≤∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ωω✳ ✳ ❊①❛♠♣❧❡✿ ▲❛✈❡r ❢♦r❝✐♥❣✱ ❍❡❝❤❧❡r ❢♦r❝✐♥❣✳ ▼✐❧❧❡r ❛♥❞ ❈♦❤❡♥ ❢♦r❝✐♥❣ ❞♦ ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❚❤❡♦r❡♠ ✭❈❛♥❥❛r ✶✾✽✽✮

d = c ✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ✉❧tr❛✜❧t❡r F s✉❝❤ t❤❛t MF ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✷

❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮

❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ MF ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ✳ ❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ✲❈❛♥❥❛r ✐❢ ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ ✳

✼ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s

❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ωω s✉❝❤ t❤❛t y ≤∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ωω✳ ✳ ❊①❛♠♣❧❡✿ ▲❛✈❡r ❢♦r❝✐♥❣✱ ❍❡❝❤❧❡r ❢♦r❝✐♥❣✳ ▼✐❧❧❡r ❛♥❞ ❈♦❤❡♥ ❢♦r❝✐♥❣ ❞♦ ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❚❤❡♦r❡♠ ✭❈❛♥❥❛r ✶✾✽✽✮

d = c ✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ✉❧tr❛✜❧t❡r F s✉❝❤ t❤❛t MF ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✷

❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮

❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ MF ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t B ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ωω✳ ❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ✲❈❛♥❥❛r ✐❢ ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ ✳

✼ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s

❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ωω s✉❝❤ t❤❛t y ≤∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ωω✳ ✳ ❊①❛♠♣❧❡✿ ▲❛✈❡r ❢♦r❝✐♥❣✱ ❍❡❝❤❧❡r ❢♦r❝✐♥❣✳ ▼✐❧❧❡r ❛♥❞ ❈♦❤❡♥ ❢♦r❝✐♥❣ ❞♦ ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❚❤❡♦r❡♠ ✭❈❛♥❥❛r ✶✾✽✽✮

d = c ✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ✉❧tr❛✜❧t❡r F s✉❝❤ t❤❛t MF ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✷

❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮

❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ MF ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t B ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ωω✳ ❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ B✲❈❛♥❥❛r ✐❢ MF ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ B✳ ✷

✼ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s

❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ωω s✉❝❤ t❤❛t y ≤∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ωω✳ ✳ ❊①❛♠♣❧❡✿ ▲❛✈❡r ❢♦r❝✐♥❣✱ ❍❡❝❤❧❡r ❢♦r❝✐♥❣✳ ▼✐❧❧❡r ❛♥❞ ❈♦❤❡♥ ❢♦r❝✐♥❣ ❞♦ ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳

❚❤❡♦r❡♠ ✭❈❛♥❥❛r ✶✾✽✽✮

d = c ✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ✉❧tr❛✜❧t❡r F s✉❝❤ t❤❛t MF ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✷

❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮

❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ MF ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t B ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ωω✳ ❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ B✲❈❛♥❥❛r ✐❢ MF ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ B✳ ✷

✼ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥

❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮

✶✮ ❊✈❡r② σ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ ✳ ▲❡t ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✱ ✐♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ ♠❛♥② ❈♦❤❡♥s✱ t❤❡r❡ ❡①✐sts ❛ ✲❈❛♥❥❛r ✳

❚❤❡♦r❡♠ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③ ✷✵✶✸❀ ❇❧❛ss✲❍r✉s❛❦✲❱❡r♥❡r ✷✵✶✶ ❢♦r ✉❧tr❛✜❧t❡rs✮

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ ❝♦❤❡r❡♥t str♦♥❣ ✲✜❧t❡r✳ ✐s ❛ str♦♥❣ ✲✜❧t❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs s✉❝❤ t❤❛t ✐❢ ❢♦r ❛❧❧ ✱ t❤❡♥ ✳

✽ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥

❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮

✶✮ ❊✈❡r② σ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ (b = c)✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✱ ✐♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ ♠❛♥② ❈♦❤❡♥s✱ t❤❡r❡ ❡①✐sts ❛ ✲❈❛♥❥❛r ✳

❚❤❡♦r❡♠ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③ ✷✵✶✸❀ ❇❧❛ss✲❍r✉s❛❦✲❱❡r♥❡r ✷✵✶✶ ❢♦r ✉❧tr❛✜❧t❡rs✮

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ ❝♦❤❡r❡♥t str♦♥❣ ✲✜❧t❡r✳ ✐s ❛ str♦♥❣ ✲✜❧t❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs s✉❝❤ t❤❛t ✐❢ ❢♦r ❛❧❧ ✱ t❤❡♥ ✳

✽ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥

❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮

✶✮ ❊✈❡r② σ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ (b = c)✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bα ≤∗ bβ ❢♦r ❛❧❧ α < β✱ ✐♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ ♠❛♥② ❈♦❤❡♥s✱ t❤❡r❡ ❡①✐sts ❛ ✲❈❛♥❥❛r ✳

❚❤❡♦r❡♠ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③ ✷✵✶✸❀ ❇❧❛ss✲❍r✉s❛❦✲❱❡r♥❡r ✷✵✶✶ ❢♦r ✉❧tr❛✜❧t❡rs✮

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ ❝♦❤❡r❡♥t str♦♥❣ ✲✜❧t❡r✳ ✐s ❛ str♦♥❣ ✲✜❧t❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs s✉❝❤ t❤❛t ✐❢ ❢♦r ❛❧❧ ✱ t❤❡♥ ✳

✽ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥

❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮

✶✮ ❊✈❡r② σ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ (b = c)✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bα ≤∗ bβ ❢♦r ❛❧❧ α < β✱ ✐♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ b ♠❛♥② ❈♦❤❡♥s✱ t❤❡r❡ ❡①✐sts ❛ B✲❈❛♥❥❛r F ⊃ FA✳ ✷

❚❤❡♦r❡♠ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③ ✷✵✶✸❀ ❇❧❛ss✲❍r✉s❛❦✲❱❡r♥❡r ✷✵✶✶ ❢♦r ✉❧tr❛✜❧t❡rs✮

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ ❝♦❤❡r❡♥t str♦♥❣ ✲✜❧t❡r✳ ✐s ❛ str♦♥❣ ✲✜❧t❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs s✉❝❤ t❤❛t ✐❢ ❢♦r ❛❧❧ ✱ t❤❡♥ ✳

✽ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥

❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮

✶✮ ❊✈❡r② σ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ (b = c)✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bα ≤∗ bβ ❢♦r ❛❧❧ α < β✱ ✐♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ b ♠❛♥② ❈♦❤❡♥s✱ t❤❡r❡ ❡①✐sts ❛ B✲❈❛♥❥❛r F ⊃ FA✳ ✷

❚❤❡♦r❡♠ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③ ✷✵✶✸❀ ❇❧❛ss✲❍r✉s❛❦✲❱❡r♥❡r ✷✵✶✶ ❢♦r ✉❧tr❛✜❧t❡rs✮

❆ ✜❧t❡r F ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ ❝♦❤❡r❡♥t str♦♥❣ P +✲✜❧t❡r✳ ✷ ✐s ❛ str♦♥❣ ✲✜❧t❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs s✉❝❤ t❤❛t ✐❢ ❢♦r ❛❧❧ ✱ t❤❡♥ ✳

✽ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥

❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮

✶✮ ❊✈❡r② σ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ (b = c)✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bα ≤∗ bβ ❢♦r ❛❧❧ α < β✱ ✐♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ b ♠❛♥② ❈♦❤❡♥s✱ t❤❡r❡ ❡①✐sts ❛ B✲❈❛♥❥❛r F ⊃ FA✳ ✷

❚❤❡♦r❡♠ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③ ✷✵✶✸❀ ❇❧❛ss✲❍r✉s❛❦✲❱❡r♥❡r ✷✵✶✶ ❢♦r ✉❧tr❛✜❧t❡rs✮

❆ ✜❧t❡r F ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ ❝♦❤❡r❡♥t str♦♥❣ P +✲✜❧t❡r✳ ✷ F ✐s ❛ str♦♥❣ P +✲✜❧t❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Cn : n ∈ ω ♦❢ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ F+ t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ kn : n ∈ ω ♦❢ ✐♥t❡❣❡rs s✉❝❤ t❤❛t ✐❢ Xn ∈ Cn ❢♦r ❛❧❧ n ✱ t❤❡♥ ✳

✽ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥

❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮

✶✮ ❊✈❡r② σ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ (b = c)✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bα ≤∗ bβ ❢♦r ❛❧❧ α < β✱ ✐♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ b ♠❛♥② ❈♦❤❡♥s✱ t❤❡r❡ ❡①✐sts ❛ B✲❈❛♥❥❛r F ⊃ FA✳ ✷

❚❤❡♦r❡♠ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③ ✷✵✶✸❀ ❇❧❛ss✲❍r✉s❛❦✲❱❡r♥❡r ✷✵✶✶ ❢♦r ✉❧tr❛✜❧t❡rs✮

❆ ✜❧t❡r F ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ ❝♦❤❡r❡♥t str♦♥❣ P +✲✜❧t❡r✳ ✷ F ✐s ❛ str♦♥❣ P +✲✜❧t❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Cn : n ∈ ω ♦❢ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ F+ t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ kn : n ∈ ω ♦❢ ✐♥t❡❣❡rs s✉❝❤ t❤❛t ✐❢ Xn ∈ Cn ❢♦r ❛❧❧ n✱ t❤❡♥

• n∈ω(Xn ∩ [kn, kn+1)) ∈ F+✳

✽ ✴ ✶✽

MF ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥

❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮

✶✮ ❊✈❡r② σ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ (b = c)✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ B = {bα : α < b} ⊂ ωω s✉❝❤ t❤❛t bα ≤∗ bβ ❢♦r ❛❧❧ α < β✱ ✐♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ b ♠❛♥② ❈♦❤❡♥s✱ t❤❡r❡ ❡①✐sts ❛ B✲❈❛♥❥❛r F ⊃ FA✳ ✷

❚❤❡♦r❡♠ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③ ✷✵✶✸❀ ❇❧❛ss✲❍r✉s❛❦✲❱❡r♥❡r ✷✵✶✶ ❢♦r ✉❧tr❛✜❧t❡rs✮

❆ ✜❧t❡r F ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ ❝♦❤❡r❡♥t str♦♥❣ P +✲✜❧t❡r✳ ✷ F ✐s ❛ str♦♥❣ P +✲✜❧t❡r ✐❢ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ Cn : n ∈ ω ♦❢ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ F+ t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ kn : n ∈ ω ♦❢ ✐♥t❡❣❡rs s✉❝❤ t❤❛t ✐❢ Xn ∈ Cn ❢♦r ❛❧❧ n✱ t❤❡♥

• n∈ω(Xn ∩ [kn, kn+1)) ∈ F+✳

✽ ✴ ✶✽

❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ r❡♣❡❛t❡❞

❚❤❡♦r❡♠ ✭❍✉r❡✇✐❝③ ✶✾✷✺✮

❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐✛ f[X] ✐s ❜♦✉♥❞❡❞ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ ✐s ▼❡♥❣❡r ✐✛ ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s ✳

✾ ✴ ✶✽

❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ r❡♣❡❛t❡❞

❚❤❡♦r❡♠ ✭❍✉r❡✇✐❝③ ✶✾✷✺✮

❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐✛ f[X] ✐s ❜♦✉♥❞❡❞ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f[X] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳

✾ ✴ ✶✽

❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✱ r❡♣❡❛t❡❞

❚❤❡♦r❡♠ ✭❍✉r❡✇✐❝③ ✶✾✷✺✮

❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ❍✉r❡✇✐❝③ ✐✛ f[X] ✐s ❜♦✉♥❞❡❞ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f[X] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ωω✳

✾ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ ✐s ❈❛♥❥❛r ✐✛ ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ ✳ ✷✮ ✐s ❛❧♠♦st ✲❜♦✉♥❞✐♥❣ ✐✛ ✐s ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ ✐✛ ✐s ❍✉r❡✇✐❝③✳

❈♦r♦❧❧❛r②

▲❡t ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ✳ ❚❤❡♥ ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ ✐s ✲❝♦♠♣❛❝t✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② ♦♥ s✉❝❤ t❤❛t ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ ✐s ♥♦t ❈❛♥❥❛r✮✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ ✐s ❈❛♥❥❛r ✐✛ ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ ✳ ✷✮ ✐s ❛❧♠♦st ✲❜♦✉♥❞✐♥❣ ✐✛ ✐s ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ ✐✛ ✐s ❍✉r❡✇✐❝③✳

❈♦r♦❧❧❛r②

▲❡t ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ✳ ❚❤❡♥ ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ ✐s ✲❝♦♠♣❛❝t✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② ♦♥ s✉❝❤ t❤❛t ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ ✐s ♥♦t ❈❛♥❥❛r✮✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ P(ω)✳ ✷✮ ✐s ❛❧♠♦st ✲❜♦✉♥❞✐♥❣ ✐✛ ✐s ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ ✐✛ ✐s ❍✉r❡✇✐❝③✳

❈♦r♦❧❧❛r②

▲❡t ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ✳ ❚❤❡♥ ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ ✐s ✲❝♦♠♣❛❝t✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② ♦♥ s✉❝❤ t❤❛t ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ ✐s ♥♦t ❈❛♥❥❛r✮✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ P(ω)✳ ✷✮ MF ✐s ❛❧♠♦st ωω✲❜♦✉♥❞✐♥❣ ✐✛ ✐s ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ ✐✛ ✐s ❍✉r❡✇✐❝③✳

❈♦r♦❧❧❛r②

▲❡t ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ✳ ❚❤❡♥ ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ ✐s ✲❝♦♠♣❛❝t✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② ♦♥ s✉❝❤ t❤❛t ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ ✐s ♥♦t ❈❛♥❥❛r✮✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ P(ω)✳ ✷✮ MF ✐s ❛❧♠♦st ωω✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s B✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ωω ✐✛ ✐s ❍✉r❡✇✐❝③✳

❈♦r♦❧❧❛r②

▲❡t ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ✳ ❚❤❡♥ ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ ✐s ✲❝♦♠♣❛❝t✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② ♦♥ s✉❝❤ t❤❛t ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ ✐s ♥♦t ❈❛♥❥❛r✮✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ P(ω)✳ ✷✮ MF ✐s ❛❧♠♦st ωω✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s B✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ωω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷

❈♦r♦❧❧❛r②

▲❡t ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ✳ ❚❤❡♥ ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ ✐s ✲❝♦♠♣❛❝t✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② ♦♥ s✉❝❤ t❤❛t ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ ✐s ♥♦t ❈❛♥❥❛r✮✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ P(ω)✳ ✷✮ MF ✐s ❛❧♠♦st ωω✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s B✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ωω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷

❈♦r♦❧❧❛r②

▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω✳ ❚❤❡♥ ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ ✐s ✲❝♦♠♣❛❝t✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② ♦♥ s✉❝❤ t❤❛t ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ ✐s ♥♦t ❈❛♥❥❛r✮✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ P(ω)✳ ✷✮ MF ✐s ❛❧♠♦st ωω✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s B✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ωω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷

❈♦r♦❧❧❛r②

▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω✳ ❚❤❡♥ MF ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ F ✐s σ✲❝♦♠♣❛❝t✳ ✷ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② ♦♥ s✉❝❤ t❤❛t ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ ✐s ♥♦t ❈❛♥❥❛r✮✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ P(ω)✳ ✷✮ MF ✐s ❛❧♠♦st ωω✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s B✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ωω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷

❈♦r♦❧❧❛r②

▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω✳ ❚❤❡♥ MF ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ F ✐s σ✲❝♦♠♣❛❝t✳ ✷ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② ♦♥ s✉❝❤ t❤❛t ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ ✐s ♥♦t ❈❛♥❥❛r✮✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ P(ω)✳ ✷✮ MF ✐s ❛❧♠♦st ωω✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s B✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ωω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷

❈♦r♦❧❧❛r②

▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω✳ ❚❤❡♥ MF ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ F ✐s σ✲❝♦♠♣❛❝t✳ ✷ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② ♦♥ s✉❝❤ t❤❛t ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ ✐s ♥♦t ❈❛♥❥❛r✮✳ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ P(ω)✳ ✷✮ MF ✐s ❛❧♠♦st ωω✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s B✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ωω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷

❈♦r♦❧❧❛r②

▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω✳ ❚❤❡♥ MF ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ F ✐s σ✲❝♦♠♣❛❝t✳ ✷ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② A ♦♥ ω s✉❝❤ t❤❛t MF(A) ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ F(A) ✐s ♥♦t ❈❛♥❥❛r✮✳ ✷ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ P(ω)✳ ✷✮ MF ✐s ❛❧♠♦st ωω✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s B✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ωω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷

❈♦r♦❧❧❛r②

▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω✳ ❚❤❡♥ MF ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ F ✐s σ✲❝♦♠♣❛❝t✳ ✷ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② A ♦♥ ω s✉❝❤ t❤❛t MF(A) ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ F(A) ✐s ♥♦t ❈❛♥❥❛r✮✳ ✷ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ ✲✜❧t❡r✳

✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮

❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮

✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ s✉❜s♣❛❝❡ ♦❢ P(ω)✳ ✷✮ MF ✐s ❛❧♠♦st ωω✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s B✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ωω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷

❈♦r♦❧❧❛r②

▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω✳ ❚❤❡♥ MF ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ F ✐s σ✲❝♦♠♣❛❝t✳ ✷ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ❋♦r ❇♦r❡❧ ✜❧t❡rs ❤❛s ❜❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ♣r♦✈❡❞ ❜② ●✉③♠❛♥✱ ❍r✉s❛❦✱ ❛♥❞ ▼❛rt✐♥❡③✳

❈♦r♦❧❧❛r② ✭❍r✉➨á❦✲▼❛rtí♥❡③ ✷✵✶✷✮

❚❤❡r❡ ❡①✐sts ❛ ♠❛❞ ❢❛♠✐❧② A ♦♥ ω s✉❝❤ t❤❛t MF(A) ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ F(A) ✐s ♥♦t ❈❛♥❥❛r✮✳ ✷ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳

❈♦r♦❧❧❛r②

❆ ✜❧t❡r F ✐s ❈❛♥❥❛r ✐✛ ✐t ✐s ❛ str♦♥❣ P +✲✜❧t❡r✳ ✷ ✶✵ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D

❆ s♣❛❝❡ (X, τ) ✐s ❝❛❧❧❡❞ ❛ D✲s♣❛❝❡✱ ✐❢ ❢♦r ❡✈❡r② f : X → τ s✉❝❤ t❤❛t x ∈ f(x) ❢♦r ❛❧❧ x✱ t❤❡r❡ ❡①✐sts ❛ ❝❧♦s❡❞ ❞✐s❝r❡t❡ D ⊂ X s✉❝❤ t❤❛t X =

x∈D f(x)✳

Pr♦❜❧❡♠

■s ❡✈❡r② r❡❣✉❧❛r ▲✐♥❞❡❧ö❢ s♣❛❝❡ ❛ ✲s♣❛❝❡❄

❚❤❡♦r❡♠ ✭❆✉r✐❝❤✐ ✷✵✶✵✮

▼❡♥❣❡r s♣❛❝❡s ❛r❡ ✲s♣❛❝❡s✳

❚❤❡♦r❡♠ ✭❊ss❡♥t✐❛❧❧② ❆✳ ❉♦✇✮

▲❡t ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤❡♥ ✐s ▼❡♥❣❡r ✐♥ ✳ Pr♦♦❢✳ ❚✇♦ st❡♣s✳ ✶✳ r❡♠❛✐♥s ▲✐♥❞❡❧ö❢✳ ✷✳ ❜❡❝♦♠❡s ▼❡♥❣❡r✳

✶✶ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D

❆ s♣❛❝❡ (X, τ) ✐s ❝❛❧❧❡❞ ❛ D✲s♣❛❝❡✱ ✐❢ ❢♦r ❡✈❡r② f : X → τ s✉❝❤ t❤❛t x ∈ f(x) ❢♦r ❛❧❧ x✱ t❤❡r❡ ❡①✐sts ❛ ❝❧♦s❡❞ ❞✐s❝r❡t❡ D ⊂ X s✉❝❤ t❤❛t X =

x∈D f(x)✳

Pr♦❜❧❡♠

■s ❡✈❡r② r❡❣✉❧❛r ▲✐♥❞❡❧ö❢ s♣❛❝❡ ❛ D✲s♣❛❝❡❄

❚❤❡♦r❡♠ ✭❆✉r✐❝❤✐ ✷✵✶✵✮

▼❡♥❣❡r s♣❛❝❡s ❛r❡ ✲s♣❛❝❡s✳

❚❤❡♦r❡♠ ✭❊ss❡♥t✐❛❧❧② ❆✳ ❉♦✇✮

▲❡t ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤❡♥ ✐s ▼❡♥❣❡r ✐♥ ✳ Pr♦♦❢✳ ❚✇♦ st❡♣s✳ ✶✳ r❡♠❛✐♥s ▲✐♥❞❡❧ö❢✳ ✷✳ ❜❡❝♦♠❡s ▼❡♥❣❡r✳

✶✶ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D

❆ s♣❛❝❡ (X, τ) ✐s ❝❛❧❧❡❞ ❛ D✲s♣❛❝❡✱ ✐❢ ❢♦r ❡✈❡r② f : X → τ s✉❝❤ t❤❛t x ∈ f(x) ❢♦r ❛❧❧ x✱ t❤❡r❡ ❡①✐sts ❛ ❝❧♦s❡❞ ❞✐s❝r❡t❡ D ⊂ X s✉❝❤ t❤❛t X =

x∈D f(x)✳

Pr♦❜❧❡♠

■s ❡✈❡r② r❡❣✉❧❛r ▲✐♥❞❡❧ö❢ s♣❛❝❡ ❛ D✲s♣❛❝❡❄

❚❤❡♦r❡♠ ✭❆✉r✐❝❤✐ ✷✵✶✵✮

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D✲s♣❛❝❡s✳

❚❤❡♦r❡♠ ✭❊ss❡♥t✐❛❧❧② ❆✳ ❉♦✇✮

▲❡t ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤❡♥ ✐s ▼❡♥❣❡r ✐♥ ✳ Pr♦♦❢✳ ❚✇♦ st❡♣s✳ ✶✳ r❡♠❛✐♥s ▲✐♥❞❡❧ö❢✳ ✷✳ ❜❡❝♦♠❡s ▼❡♥❣❡r✳

✶✶ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D

❆ s♣❛❝❡ (X, τ) ✐s ❝❛❧❧❡❞ ❛ D✲s♣❛❝❡✱ ✐❢ ❢♦r ❡✈❡r② f : X → τ s✉❝❤ t❤❛t x ∈ f(x) ❢♦r ❛❧❧ x✱ t❤❡r❡ ❡①✐sts ❛ ❝❧♦s❡❞ ❞✐s❝r❡t❡ D ⊂ X s✉❝❤ t❤❛t X =

x∈D f(x)✳

Pr♦❜❧❡♠

■s ❡✈❡r② r❡❣✉❧❛r ▲✐♥❞❡❧ö❢ s♣❛❝❡ ❛ D✲s♣❛❝❡❄

❚❤❡♦r❡♠ ✭❆✉r✐❝❤✐ ✷✵✶✵✮

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D✲s♣❛❝❡s✳

❚❤❡♦r❡♠ ✭❊ss❡♥t✐❛❧❧② ❆✳ ❉♦✇✮

▲❡t (X, τ) ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤❡♥ X ✐s ▼❡♥❣❡r ✐♥ V Fn(µ,2)✳ Pr♦♦❢✳ ❚✇♦ st❡♣s✳ ✶✳ r❡♠❛✐♥s ▲✐♥❞❡❧ö❢✳ ✷✳ ❜❡❝♦♠❡s ▼❡♥❣❡r✳

✶✶ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D

❆ s♣❛❝❡ (X, τ) ✐s ❝❛❧❧❡❞ ❛ D✲s♣❛❝❡✱ ✐❢ ❢♦r ❡✈❡r② f : X → τ s✉❝❤ t❤❛t x ∈ f(x) ❢♦r ❛❧❧ x✱ t❤❡r❡ ❡①✐sts ❛ ❝❧♦s❡❞ ❞✐s❝r❡t❡ D ⊂ X s✉❝❤ t❤❛t X =

x∈D f(x)✳

Pr♦❜❧❡♠

■s ❡✈❡r② r❡❣✉❧❛r ▲✐♥❞❡❧ö❢ s♣❛❝❡ ❛ D✲s♣❛❝❡❄

❚❤❡♦r❡♠ ✭❆✉r✐❝❤✐ ✷✵✶✵✮

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D✲s♣❛❝❡s✳

❚❤❡♦r❡♠ ✭❊ss❡♥t✐❛❧❧② ❆✳ ❉♦✇✮

▲❡t (X, τ) ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤❡♥ X ✐s ▼❡♥❣❡r ✐♥ V Fn(µ,2)✳ Pr♦♦❢✳ ❚✇♦ st❡♣s✳ ✶✳ X r❡♠❛✐♥s ▲✐♥❞❡❧ö❢✳ ✷✳ ❜❡❝♦♠❡s ▼❡♥❣❡r✳

✶✶ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D

❆ s♣❛❝❡ (X, τ) ✐s ❝❛❧❧❡❞ ❛ D✲s♣❛❝❡✱ ✐❢ ❢♦r ❡✈❡r② f : X → τ s✉❝❤ t❤❛t x ∈ f(x) ❢♦r ❛❧❧ x✱ t❤❡r❡ ❡①✐sts ❛ ❝❧♦s❡❞ ❞✐s❝r❡t❡ D ⊂ X s✉❝❤ t❤❛t X =

x∈D f(x)✳

Pr♦❜❧❡♠

■s ❡✈❡r② r❡❣✉❧❛r ▲✐♥❞❡❧ö❢ s♣❛❝❡ ❛ D✲s♣❛❝❡❄

❚❤❡♦r❡♠ ✭❆✉r✐❝❤✐ ✷✵✶✵✮

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D✲s♣❛❝❡s✳

❚❤❡♦r❡♠ ✭❊ss❡♥t✐❛❧❧② ❆✳ ❉♦✇✮

▲❡t (X, τ) ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤❡♥ X ✐s ▼❡♥❣❡r ✐♥ V Fn(µ,2)✳ Pr♦♦❢✳ ❚✇♦ st❡♣s✳ ✶✳ X r❡♠❛✐♥s ▲✐♥❞❡❧ö❢✳ ✷✳ X ❜❡❝♦♠❡s ▼❡♥❣❡r✳

✶✶ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D

❆ s♣❛❝❡ (X, τ) ✐s ❝❛❧❧❡❞ ❛ D✲s♣❛❝❡✱ ✐❢ ❢♦r ❡✈❡r② f : X → τ s✉❝❤ t❤❛t x ∈ f(x) ❢♦r ❛❧❧ x✱ t❤❡r❡ ❡①✐sts ❛ ❝❧♦s❡❞ ❞✐s❝r❡t❡ D ⊂ X s✉❝❤ t❤❛t X =

x∈D f(x)✳

Pr♦❜❧❡♠

■s ❡✈❡r② r❡❣✉❧❛r ▲✐♥❞❡❧ö❢ s♣❛❝❡ ❛ D✲s♣❛❝❡❄

❚❤❡♦r❡♠ ✭❆✉r✐❝❤✐ ✷✵✶✵✮

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D✲s♣❛❝❡s✳

❚❤❡♦r❡♠ ✭❊ss❡♥t✐❛❧❧② ❆✳ ❉♦✇✮

▲❡t (X, τ) ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤❡♥ X ✐s ▼❡♥❣❡r ✐♥ V Fn(µ,2)✳ Pr♦♦❢✳ ❚✇♦ st❡♣s✳ ✶✳ X r❡♠❛✐♥s ▲✐♥❞❡❧ö❢✳ ✷✳ X ❜❡❝♦♠❡s ▼❡♥❣❡r✳ ✷

✶✶ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D

❆ s♣❛❝❡ (X, τ) ✐s ❝❛❧❧❡❞ ❛ D✲s♣❛❝❡✱ ✐❢ ❢♦r ❡✈❡r② f : X → τ s✉❝❤ t❤❛t x ∈ f(x) ❢♦r ❛❧❧ x✱ t❤❡r❡ ❡①✐sts ❛ ❝❧♦s❡❞ ❞✐s❝r❡t❡ D ⊂ X s✉❝❤ t❤❛t X =

x∈D f(x)✳

Pr♦❜❧❡♠

■s ❡✈❡r② r❡❣✉❧❛r ▲✐♥❞❡❧ö❢ s♣❛❝❡ ❛ D✲s♣❛❝❡❄

❚❤❡♦r❡♠ ✭❆✉r✐❝❤✐ ✷✵✶✵✮

▼❡♥❣❡r s♣❛❝❡s ❛r❡ D✲s♣❛❝❡s✳

❚❤❡♦r❡♠ ✭❊ss❡♥t✐❛❧❧② ❆✳ ❉♦✇✮

▲❡t (X, τ) ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤❡♥ X ✐s ▼❡♥❣❡r ✐♥ V Fn(µ,2)✳ Pr♦♦❢✳ ❚✇♦ st❡♣s✳ ✶✳ X r❡♠❛✐♥s ▲✐♥❞❡❧ö❢✳ ✷✳ X ❜❡❝♦♠❡s ▼❡♥❣❡r✳ ✷

✶✶ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛♥❞ ❚✉❦❡② ♦r❞❡r

▲❡t P✱ Q ❜❡ ❞✐r❡❝t❡❞ ♣♦s❡ts✳ ❚❤❡♥ ✐❢ t❤❡r❡ ✐s ❛ ♠❛♣ t❤❛t t❛❦❡s ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ t♦ ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ ✳

❚❤❡♦r❡♠ ✭❈❤r✐st❡♥s❡♥ ✶✾✼✹✮

■❢ ✐s ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✱ t❤❡♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s P♦❧✐s❤✳

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❛♠❛t❡❧❛s❤✈✐❧✐ ✷✵✶❄✮

▲❡t ❜❡ ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✳ ❚❤❡♥ ✐✛ ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡✳

◗✉❡st✐♦♥

■s t❤❡r❡ ❛ ❩❋❈ ❡①❛♠♣❧❡ ♦❢ ❛ s♣❛❝❡ s✉❝❤ t❤❛t ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮

▲❡t ❜❡ ▼❡♥❣❡r ♥♦♥✲ ✲❝♦♠♣❛❝t✳ ❚❤❡♥ ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤✳

✶✷ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛♥❞ ❚✉❦❡② ♦r❞❡r

▲❡t P✱ Q ❜❡ ❞✐r❡❝t❡❞ ♣♦s❡ts✳ ❚❤❡♥ P ≥T Q ✐❢ t❤❡r❡ ✐s ❛ ♠❛♣ φ : P → Q t❤❛t t❛❦❡s ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ P t♦ ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ Q✳

❚❤❡♦r❡♠ ✭❈❤r✐st❡♥s❡♥ ✶✾✼✹✮

■❢ ✐s ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✱ t❤❡♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s P♦❧✐s❤✳

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❛♠❛t❡❧❛s❤✈✐❧✐ ✷✵✶❄✮

▲❡t ❜❡ ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✳ ❚❤❡♥ ✐✛ ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡✳

◗✉❡st✐♦♥

■s t❤❡r❡ ❛ ❩❋❈ ❡①❛♠♣❧❡ ♦❢ ❛ s♣❛❝❡ s✉❝❤ t❤❛t ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮

▲❡t ❜❡ ▼❡♥❣❡r ♥♦♥✲ ✲❝♦♠♣❛❝t✳ ❚❤❡♥ ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤✳

✶✷ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛♥❞ ❚✉❦❡② ♦r❞❡r

▲❡t P✱ Q ❜❡ ❞✐r❡❝t❡❞ ♣♦s❡ts✳ ❚❤❡♥ P ≥T Q ✐❢ t❤❡r❡ ✐s ❛ ♠❛♣ φ : P → Q t❤❛t t❛❦❡s ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ P t♦ ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ Q✳

❚❤❡♦r❡♠ ✭❈❤r✐st❡♥s❡♥ ✶✾✼✹✮

■❢ M ✐s ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✱ t❤❡♥ ωω ≥T K(M) ✐❢ ❛♥❞ ♦♥❧② ✐❢ M ✐s P♦❧✐s❤✳

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❛♠❛t❡❧❛s❤✈✐❧✐ ✷✵✶❄✮

▲❡t ❜❡ ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✳ ❚❤❡♥ ✐✛ ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡✳

◗✉❡st✐♦♥

■s t❤❡r❡ ❛ ❩❋❈ ❡①❛♠♣❧❡ ♦❢ ❛ s♣❛❝❡ s✉❝❤ t❤❛t ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮

▲❡t ❜❡ ▼❡♥❣❡r ♥♦♥✲ ✲❝♦♠♣❛❝t✳ ❚❤❡♥ ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤✳

✶✷ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛♥❞ ❚✉❦❡② ♦r❞❡r

▲❡t P✱ Q ❜❡ ❞✐r❡❝t❡❞ ♣♦s❡ts✳ ❚❤❡♥ P ≥T Q ✐❢ t❤❡r❡ ✐s ❛ ♠❛♣ φ : P → Q t❤❛t t❛❦❡s ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ P t♦ ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ Q✳

❚❤❡♦r❡♠ ✭❈❤r✐st❡♥s❡♥ ✶✾✼✹✮

■❢ M ✐s ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✱ t❤❡♥ ωω ≥T K(M) ✐❢ ❛♥❞ ♦♥❧② ✐❢ M ✐s P♦❧✐s❤✳

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❛♠❛t❡❧❛s❤✈✐❧✐ ✷✵✶❄✮

▲❡t M ❜❡ ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✳ ❚❤❡♥ K(M) ≥T K(Q) ✐✛ K(M) ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡✳

◗✉❡st✐♦♥

■s t❤❡r❡ ❛ ❩❋❈ ❡①❛♠♣❧❡ ♦❢ ❛ s♣❛❝❡ s✉❝❤ t❤❛t ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮

▲❡t ❜❡ ▼❡♥❣❡r ♥♦♥✲ ✲❝♦♠♣❛❝t✳ ❚❤❡♥ ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤✳

✶✷ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛♥❞ ❚✉❦❡② ♦r❞❡r

▲❡t P✱ Q ❜❡ ❞✐r❡❝t❡❞ ♣♦s❡ts✳ ❚❤❡♥ P ≥T Q ✐❢ t❤❡r❡ ✐s ❛ ♠❛♣ φ : P → Q t❤❛t t❛❦❡s ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ P t♦ ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ Q✳

❚❤❡♦r❡♠ ✭❈❤r✐st❡♥s❡♥ ✶✾✼✹✮

■❢ M ✐s ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✱ t❤❡♥ ωω ≥T K(M) ✐❢ ❛♥❞ ♦♥❧② ✐❢ M ✐s P♦❧✐s❤✳

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❛♠❛t❡❧❛s❤✈✐❧✐ ✷✵✶❄✮

▲❡t M ❜❡ ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✳ ❚❤❡♥ K(M) ≥T K(Q) ✐✛ K(M) ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡✳

◗✉❡st✐♦♥

■s t❤❡r❡ ❛ ❩❋❈ ❡①❛♠♣❧❡ ♦❢ ❛ s♣❛❝❡ M s✉❝❤ t❤❛t K(M) ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮

▲❡t ❜❡ ▼❡♥❣❡r ♥♦♥✲ ✲❝♦♠♣❛❝t✳ ❚❤❡♥ ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤✳

✶✷ ✴ ✶✽

▼❡♥❣❡r s♣❛❝❡s ❛♥❞ ❚✉❦❡② ♦r❞❡r

▲❡t P✱ Q ❜❡ ❞✐r❡❝t❡❞ ♣♦s❡ts✳ ❚❤❡♥ P ≥T Q ✐❢ t❤❡r❡ ✐s ❛ ♠❛♣ φ : P → Q t❤❛t t❛❦❡s ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ P t♦ ❝♦✜♥❛❧ s✉❜s❡ts ♦❢ Q✳

❚❤❡♦r❡♠ ✭❈❤r✐st❡♥s❡♥ ✶✾✼✹✮

■❢ M ✐s ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✱ t❤❡♥ ωω ≥T K(M) ✐❢ ❛♥❞ ♦♥❧② ✐❢ M ✐s P♦❧✐s❤✳

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❛♠❛t❡❧❛s❤✈✐❧✐ ✷✵✶❄✮

▲❡t M ❜❡ ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐③❛❜❧❡ s♣❛❝❡✳ ❚❤❡♥ K(M) ≥T K(Q) ✐✛ K(M) ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡✳

◗✉❡st✐♦♥

■s t❤❡r❡ ❛ ❩❋❈ ❡①❛♠♣❧❡ ♦❢ ❛ s♣❛❝❡ M s✉❝❤ t❤❛t K(M) ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮

▲❡t X ⊂ 2ω ❜❡ ▼❡♥❣❡r ♥♦♥✲σ✲❝♦♠♣❛❝t✳ ❚❤❡♥ K(2ω \ X) ✐s ❤❡r❡❞✐t❛r✐❧② ❇❛✐r❡ ♥♦♥✲P♦❧✐s❤✳

✶✷ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ ❚✉❦❡② ♦r❞❡r

◗✉❡st✐♦♥ ✭❋r❡♠❧✐♥ ✶✾✾✶✮

■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ X ⊂ 2ω s✉❝❤ t❤❛t K(Q) ≥T K(X)❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮

✭❱❂▲✮✳ ❚❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ s✉❝❤ t❤❛t ✳ ❲❡ ❝♦♥str✉❝t ❛ ❝♦✲❛♥❛❧②t✐❝ ❍✉r❡✇✐❝③ s✉❝❤ t❤❛t ✐s ❛s r❡q✉✐r❡❞✳ ❲❡ ✉s❡ r❡s✉❧ts ♦❢ ❱✐❞♥②❛♥s③❦② t♦ ♠❛❦❡ s✉r❡ t❤❛t ✐s ❝♦✲❛♥❛❧②t✐❝✱ ✇❤✐❝❤ ❡①t❡♥❞ ❛♥❞ ✉♥✐❢② ❡❛r❧✐❡r r❡s✉❧ts ♦❢ ❆✳ ▼✐❧❧❡r✳

✶✸ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ ❚✉❦❡② ♦r❞❡r

◗✉❡st✐♦♥ ✭❋r❡♠❧✐♥ ✶✾✾✶✮

■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ X ⊂ 2ω s✉❝❤ t❤❛t K(Q) ≥T K(X)❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮

✭❱❂▲✮✳ ❚❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ X ⊂ 2ω s✉❝❤ t❤❛t K(Q) >T K(X)✳ ❲❡ ❝♦♥str✉❝t ❛ ❝♦✲❛♥❛❧②t✐❝ ❍✉r❡✇✐❝③ s✉❝❤ t❤❛t ✐s ❛s r❡q✉✐r❡❞✳ ❲❡ ✉s❡ r❡s✉❧ts ♦❢ ❱✐❞♥②❛♥s③❦② t♦ ♠❛❦❡ s✉r❡ t❤❛t ✐s ❝♦✲❛♥❛❧②t✐❝✱ ✇❤✐❝❤ ❡①t❡♥❞ ❛♥❞ ✉♥✐❢② ❡❛r❧✐❡r r❡s✉❧ts ♦❢ ❆✳ ▼✐❧❧❡r✳

✶✸ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ ❚✉❦❡② ♦r❞❡r

◗✉❡st✐♦♥ ✭❋r❡♠❧✐♥ ✶✾✾✶✮

■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ X ⊂ 2ω s✉❝❤ t❤❛t K(Q) ≥T K(X)❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮

✭❱❂▲✮✳ ❚❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ X ⊂ 2ω s✉❝❤ t❤❛t K(Q) >T K(X)✳ ❲❡ ❝♦♥str✉❝t ❛ ❝♦✲❛♥❛❧②t✐❝ ❍✉r❡✇✐❝③ Y ⊂ 2ω s✉❝❤ t❤❛t X = 2ω \ Y ✐s ❛s r❡q✉✐r❡❞✳ ❲❡ ✉s❡ r❡s✉❧ts ♦❢ ❱✐❞♥②❛♥s③❦② t♦ ♠❛❦❡ s✉r❡ t❤❛t ✐s ❝♦✲❛♥❛❧②t✐❝✱ ✇❤✐❝❤ ❡①t❡♥❞ ❛♥❞ ✉♥✐❢② ❡❛r❧✐❡r r❡s✉❧ts ♦❢ ❆✳ ▼✐❧❧❡r✳

✶✸ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ ❚✉❦❡② ♦r❞❡r

◗✉❡st✐♦♥ ✭❋r❡♠❧✐♥ ✶✾✾✶✮

■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ X ⊂ 2ω s✉❝❤ t❤❛t K(Q) ≥T K(X)❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮

✭❱❂▲✮✳ ❚❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ X ⊂ 2ω s✉❝❤ t❤❛t K(Q) >T K(X)✳ ❲❡ ❝♦♥str✉❝t ❛ ❝♦✲❛♥❛❧②t✐❝ ❍✉r❡✇✐❝③ Y ⊂ 2ω s✉❝❤ t❤❛t X = 2ω \ Y ✐s ❛s r❡q✉✐r❡❞✳ ❲❡ ✉s❡ r❡s✉❧ts ♦❢ ❱✐❞♥②❛♥s③❦② t♦ ♠❛❦❡ s✉r❡ t❤❛t Y ✐s ❝♦✲❛♥❛❧②t✐❝✱ ✇❤✐❝❤ ❡①t❡♥❞ ❛♥❞ ✉♥✐❢② ❡❛r❧✐❡r r❡s✉❧ts ♦❢ ❆✳ ▼✐❧❧❡r✳

✶✸ ✴ ✶✽

Pr❡s❡r✈❛t✐♦♥ ❜② ♣r♦❞✉❝ts

❋❛❝t✳ ✭❈❍✳✮ ❚❤❡r❡ ❛r❡ t✇♦ ❙✐❡r♣✐♥s❦✐ ✭❤❡♥❝❡ ❍✉r❡✇✐❝③✮ s❡ts S0, S1 ✇❤♦s❡ ♣r♦❞✉❝t ✐s ♥♦t ▼❡♥❣❡r✳ Pr♦♦❢✳ ❋✐① ❛ ❝♦✉♥t❛❜❧❡ ❞❡♥s❡ Q ⊂ 2ω ❛♥❞ ✇r✐t❡ 2ω \ Q = {xα : α < ω1}✳ ■♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ❙✐❡r♣✐♥s❦✐ s❡t ❜② tr❛♥s✜♥✐t❡ ✐♥❞✉❝t✐♦♥ ❛t ❡❛❝❤ st❛❣❡ ✇❡ ❝❛♥ ♣✐❝❦ ❛ ♣♦✐♥t ♦✉ts✐❞❡ ♦❢ ❛ ❣✐✈❡♥ ♠❡❛s✉r❡ ③❡r♦ s❡t ✳ ❤❛s ❛ ♥❛t✉r❛❧ str✉❝t✉r❡ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣✱ ❛♥❞ t❤❡ s✉♠ ♦❢ ❛♥② t✇♦ ♠❡❛s✉r❡ ✶ s❡ts ✐s t❤❡ ✇❤♦❧❡ ❣r♦✉♣✳ ❈❤♦♦s❡ s✉❝❤ t❤❛t ❛♥❞ ✳ ❙❡t ✳

Pr♦❜❧❡♠

■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ▼❡♥❣❡r s♣❛❝❡s ✐s ▼❡♥❣❡r❄ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③ s♣❛❝❡s ✐s ❍✉r❡✇✐❝③❄ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③ s♣❛❝❡s ✐s ▼❡♥❣❡r❄

✶✹ ✴ ✶✽

Pr❡s❡r✈❛t✐♦♥ ❜② ♣r♦❞✉❝ts

❋❛❝t✳ ✭❈❍✳✮ ❚❤❡r❡ ❛r❡ t✇♦ ❙✐❡r♣✐♥s❦✐ ✭❤❡♥❝❡ ❍✉r❡✇✐❝③✮ s❡ts S0, S1 ✇❤♦s❡ ♣r♦❞✉❝t ✐s ♥♦t ▼❡♥❣❡r✳ Pr♦♦❢✳ ❋✐① ❛ ❝♦✉♥t❛❜❧❡ ❞❡♥s❡ Q ⊂ 2ω ❛♥❞ ✇r✐t❡ 2ω \ Q = {xα : α < ω1}✳ ■♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ❙✐❡r♣✐♥s❦✐ s❡t ❜② tr❛♥s✜♥✐t❡ ✐♥❞✉❝t✐♦♥ ❛t ❡❛❝❤ st❛❣❡ α ✇❡ ❝❛♥ ♣✐❝❦ ❛ ♣♦✐♥t sα ♦✉ts✐❞❡ ♦❢ ❛ ❣✐✈❡♥ ♠❡❛s✉r❡ ③❡r♦ s❡t Zα ⊂ 2ω✳ ❤❛s ❛ ♥❛t✉r❛❧ str✉❝t✉r❡ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣✱ ❛♥❞ t❤❡ s✉♠ ♦❢ ❛♥② t✇♦ ♠❡❛s✉r❡ ✶ s❡ts ✐s t❤❡ ✇❤♦❧❡ ❣r♦✉♣✳ ❈❤♦♦s❡ s✉❝❤ t❤❛t ❛♥❞ ✳ ❙❡t ✳

Pr♦❜❧❡♠

■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ▼❡♥❣❡r s♣❛❝❡s ✐s ▼❡♥❣❡r❄ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③ s♣❛❝❡s ✐s ❍✉r❡✇✐❝③❄ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③ s♣❛❝❡s ✐s ▼❡♥❣❡r❄

✶✹ ✴ ✶✽

Pr❡s❡r✈❛t✐♦♥ ❜② ♣r♦❞✉❝ts

❋❛❝t✳ ✭❈❍✳✮ ❚❤❡r❡ ❛r❡ t✇♦ ❙✐❡r♣✐♥s❦✐ ✭❤❡♥❝❡ ❍✉r❡✇✐❝③✮ s❡ts S0, S1 ✇❤♦s❡ ♣r♦❞✉❝t ✐s ♥♦t ▼❡♥❣❡r✳ Pr♦♦❢✳ ❋✐① ❛ ❝♦✉♥t❛❜❧❡ ❞❡♥s❡ Q ⊂ 2ω ❛♥❞ ✇r✐t❡ 2ω \ Q = {xα : α < ω1}✳ ■♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ❙✐❡r♣✐♥s❦✐ s❡t ❜② tr❛♥s✜♥✐t❡ ✐♥❞✉❝t✐♦♥ ❛t ❡❛❝❤ st❛❣❡ α ✇❡ ❝❛♥ ♣✐❝❦ ❛ ♣♦✐♥t sα ♦✉ts✐❞❡ ♦❢ ❛ ❣✐✈❡♥ ♠❡❛s✉r❡ ③❡r♦ s❡t Zα ⊂ 2ω✳ 2ω ❤❛s ❛ ♥❛t✉r❛❧ str✉❝t✉r❡ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣✱ ❛♥❞ t❤❡ s✉♠ ♦❢ ❛♥② t✇♦ ♠❡❛s✉r❡ ✶ s❡ts ✐s t❤❡ ✇❤♦❧❡ ❣r♦✉♣✳ ❈❤♦♦s❡ s✉❝❤ t❤❛t ❛♥❞ ✳ ❙❡t ✳

Pr♦❜❧❡♠

■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ▼❡♥❣❡r s♣❛❝❡s ✐s ▼❡♥❣❡r❄ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③ s♣❛❝❡s ✐s ❍✉r❡✇✐❝③❄ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③ s♣❛❝❡s ✐s ▼❡♥❣❡r❄

✶✹ ✴ ✶✽

Pr❡s❡r✈❛t✐♦♥ ❜② ♣r♦❞✉❝ts

❋❛❝t✳ ✭❈❍✳✮ ❚❤❡r❡ ❛r❡ t✇♦ ❙✐❡r♣✐♥s❦✐ ✭❤❡♥❝❡ ❍✉r❡✇✐❝③✮ s❡ts S0, S1 ✇❤♦s❡ ♣r♦❞✉❝t ✐s ♥♦t ▼❡♥❣❡r✳ Pr♦♦❢✳ ❋✐① ❛ ❝♦✉♥t❛❜❧❡ ❞❡♥s❡ Q ⊂ 2ω ❛♥❞ ✇r✐t❡ 2ω \ Q = {xα : α < ω1}✳ ■♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ❙✐❡r♣✐♥s❦✐ s❡t ❜② tr❛♥s✜♥✐t❡ ✐♥❞✉❝t✐♦♥ ❛t ❡❛❝❤ st❛❣❡ α ✇❡ ❝❛♥ ♣✐❝❦ ❛ ♣♦✐♥t sα ♦✉ts✐❞❡ ♦❢ ❛ ❣✐✈❡♥ ♠❡❛s✉r❡ ③❡r♦ s❡t Zα ⊂ 2ω✳ 2ω ❤❛s ❛ ♥❛t✉r❛❧ str✉❝t✉r❡ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣✱ ❛♥❞ t❤❡ s✉♠ ♦❢ ❛♥② t✇♦ ♠❡❛s✉r❡ ✶ s❡ts ✐s t❤❡ ✇❤♦❧❡ ❣r♦✉♣✳ ❈❤♦♦s❡ s0

α, s1 α ∈ 2ω \ Zα s✉❝❤ t❤❛t s0 α + s1 α = xα

❛♥❞ si

α + {s1−i β

: β < α} ∩ Q = ∅✳ ❙❡t ✳

Pr♦❜❧❡♠

■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ▼❡♥❣❡r s♣❛❝❡s ✐s ▼❡♥❣❡r❄ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③ s♣❛❝❡s ✐s ❍✉r❡✇✐❝③❄ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③ s♣❛❝❡s ✐s ▼❡♥❣❡r❄

✶✹ ✴ ✶✽

Pr❡s❡r✈❛t✐♦♥ ❜② ♣r♦❞✉❝ts

❋❛❝t✳ ✭❈❍✳✮ ❚❤❡r❡ ❛r❡ t✇♦ ❙✐❡r♣✐♥s❦✐ ✭❤❡♥❝❡ ❍✉r❡✇✐❝③✮ s❡ts S0, S1 ✇❤♦s❡ ♣r♦❞✉❝t ✐s ♥♦t ▼❡♥❣❡r✳ Pr♦♦❢✳ ❋✐① ❛ ❝♦✉♥t❛❜❧❡ ❞❡♥s❡ Q ⊂ 2ω ❛♥❞ ✇r✐t❡ 2ω \ Q = {xα : α < ω1}✳ ■♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ❙✐❡r♣✐♥s❦✐ s❡t ❜② tr❛♥s✜♥✐t❡ ✐♥❞✉❝t✐♦♥ ❛t ❡❛❝❤ st❛❣❡ α ✇❡ ❝❛♥ ♣✐❝❦ ❛ ♣♦✐♥t sα ♦✉ts✐❞❡ ♦❢ ❛ ❣✐✈❡♥ ♠❡❛s✉r❡ ③❡r♦ s❡t Zα ⊂ 2ω✳ 2ω ❤❛s ❛ ♥❛t✉r❛❧ str✉❝t✉r❡ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣✱ ❛♥❞ t❤❡ s✉♠ ♦❢ ❛♥② t✇♦ ♠❡❛s✉r❡ ✶ s❡ts ✐s t❤❡ ✇❤♦❧❡ ❣r♦✉♣✳ ❈❤♦♦s❡ s0

α, s1 α ∈ 2ω \ Zα s✉❝❤ t❤❛t s0 α + s1 α = xα

❛♥❞ si

α + {s1−i β

: β < α} ∩ Q = ∅✳ ❙❡t Si = {si

α : α < ω1}✳

✷

Pr♦❜❧❡♠

■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ▼❡♥❣❡r s♣❛❝❡s ✐s ▼❡♥❣❡r❄ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③ s♣❛❝❡s ✐s ❍✉r❡✇✐❝③❄ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③ s♣❛❝❡s ✐s ▼❡♥❣❡r❄

✶✹ ✴ ✶✽

Pr❡s❡r✈❛t✐♦♥ ❜② ♣r♦❞✉❝ts

❋❛❝t✳ ✭❈❍✳✮ ❚❤❡r❡ ❛r❡ t✇♦ ❙✐❡r♣✐♥s❦✐ ✭❤❡♥❝❡ ❍✉r❡✇✐❝③✮ s❡ts S0, S1 ✇❤♦s❡ ♣r♦❞✉❝t ✐s ♥♦t ▼❡♥❣❡r✳ Pr♦♦❢✳ ❋✐① ❛ ❝♦✉♥t❛❜❧❡ ❞❡♥s❡ Q ⊂ 2ω ❛♥❞ ✇r✐t❡ 2ω \ Q = {xα : α < ω1}✳ ■♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ❙✐❡r♣✐♥s❦✐ s❡t ❜② tr❛♥s✜♥✐t❡ ✐♥❞✉❝t✐♦♥ ❛t ❡❛❝❤ st❛❣❡ α ✇❡ ❝❛♥ ♣✐❝❦ ❛ ♣♦✐♥t sα ♦✉ts✐❞❡ ♦❢ ❛ ❣✐✈❡♥ ♠❡❛s✉r❡ ③❡r♦ s❡t Zα ⊂ 2ω✳ 2ω ❤❛s ❛ ♥❛t✉r❛❧ str✉❝t✉r❡ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣✱ ❛♥❞ t❤❡ s✉♠ ♦❢ ❛♥② t✇♦ ♠❡❛s✉r❡ ✶ s❡ts ✐s t❤❡ ✇❤♦❧❡ ❣r♦✉♣✳ ❈❤♦♦s❡ s0

α, s1 α ∈ 2ω \ Zα s✉❝❤ t❤❛t s0 α + s1 α = xα

❛♥❞ si

α + {s1−i β

: β < α} ∩ Q = ∅✳ ❙❡t Si = {si

α : α < ω1}✳

✷

Pr♦❜❧❡♠

◮ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ▼❡♥❣❡r

s♣❛❝❡s ✐s ▼❡♥❣❡r❄

◮ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③

s♣❛❝❡s ✐s ❍✉r❡✇✐❝③❄

◮ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ♠❡tr✐③❛❜❧❡ ❍✉r❡✇✐❝③

s♣❛❝❡s ✐s ▼❡♥❣❡r❄

✶✹ ✴ ✶✽

❚❤❡r❡ ❛r❡ t✇♦ r❡❛s♦♥s ✇❤② ❛ ♣r♦❞✉❝t ♦❢ ❍✉r❡✇✐❝③ s♣❛❝❡s X, Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❍✉r❡✇✐❝③✳ ♠❛② ❢❛✐❧ t♦ ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤✐s ♠❛② ✐♥❞❡❡❞ ❤❛♣♣❡♥✿ ✐♥ ❩❋❈ t❤❡r❡ ❛r❡ t✇♦ ♥♦r♠❛❧ s♣❛❝❡s ✇✐t❤ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ♠✉❝❤ str♦♥❣❡r t❤❛♥ t❤❡ ❍✉r❡✇✐❝③ ♦♥❡ s✉❝❤ t❤❛t ✐s ♥♦t ▲✐♥❞❡❧ö❢ ✭❚♦❞♦r❝❡✈✐❝ ✶✾✾✺✮✳ ✐s ▲✐♥❞❡❧ö❢✱ ❡✳❣✳✱ ✳ ❚❤✐s ❝❛s❡ ✐s s❡♥s✐t✐✈❡ t♦ t❤❡ ❛♠❜✐❡♥t s❡t✲t❤❡♦r❡t✐❝ ✉♥✐✈❡rs❡✿ ✉♥❞❡r ❈❍ t❤❡r❡ ❡①✐sts ❛ ❍✉r❡✇✐❝③ s♣❛❝❡ ✇❤♦s❡ sq✉❛r❡ ✐s ♥♦t ▼❡♥❣❡r✳ ❘❡q✉✐r❡s ❛s ♣r♦✈❡❞ ❜② ❙❝❤❡❡♣❡rs ❛♥❞ ❚s❛❜❛♥ ✐♥ ✷✵✵✷✳

❚❤❡♦r❡♠ ✭❘❡♣♦✈s✲❩✳ ✷✵✶✻✮

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❇♦r❡❧✬s ❝♦♥❥❡❝t✉r❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ ♠❡tr✐③❛❜❧❡ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt②✳ ❆s ❛ r❡s✉❧t✱ ✐♥ t❤✐s ♠♦❞❡❧ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt② ♣r♦✈✐❞❡❞ t❤❛t ✐t ✐s ▲✐♥❞❡❧ö❢ ◆♦t❡✿ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❇♦r❡❧✬s ❈♦♥❥❡❝t✉r❡✳

✶✺ ✴ ✶✽

❚❤❡r❡ ❛r❡ t✇♦ r❡❛s♦♥s ✇❤② ❛ ♣r♦❞✉❝t ♦❢ ❍✉r❡✇✐❝③ s♣❛❝❡s X, Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❍✉r❡✇✐❝③✳ X × Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤✐s ♠❛② ✐♥❞❡❡❞ ❤❛♣♣❡♥✿ ✐♥ ❩❋❈ t❤❡r❡ ❛r❡ t✇♦ ♥♦r♠❛❧ s♣❛❝❡s ✇✐t❤ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ♠✉❝❤ str♦♥❣❡r t❤❛♥ t❤❡ ❍✉r❡✇✐❝③ ♦♥❡ s✉❝❤ t❤❛t ✐s ♥♦t ▲✐♥❞❡❧ö❢ ✭❚♦❞♦r❝❡✈✐❝ ✶✾✾✺✮✳ ✐s ▲✐♥❞❡❧ö❢✱ ❡✳❣✳✱ ✳ ❚❤✐s ❝❛s❡ ✐s s❡♥s✐t✐✈❡ t♦ t❤❡ ❛♠❜✐❡♥t s❡t✲t❤❡♦r❡t✐❝ ✉♥✐✈❡rs❡✿ ✉♥❞❡r ❈❍ t❤❡r❡ ❡①✐sts ❛ ❍✉r❡✇✐❝③ s♣❛❝❡ ✇❤♦s❡ sq✉❛r❡ ✐s ♥♦t ▼❡♥❣❡r✳ ❘❡q✉✐r❡s ❛s ♣r♦✈❡❞ ❜② ❙❝❤❡❡♣❡rs ❛♥❞ ❚s❛❜❛♥ ✐♥ ✷✵✵✷✳

❚❤❡♦r❡♠ ✭❘❡♣♦✈s✲❩✳ ✷✵✶✻✮

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❇♦r❡❧✬s ❝♦♥❥❡❝t✉r❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ ♠❡tr✐③❛❜❧❡ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt②✳ ❆s ❛ r❡s✉❧t✱ ✐♥ t❤✐s ♠♦❞❡❧ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt② ♣r♦✈✐❞❡❞ t❤❛t ✐t ✐s ▲✐♥❞❡❧ö❢ ◆♦t❡✿ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❇♦r❡❧✬s ❈♦♥❥❡❝t✉r❡✳

✶✺ ✴ ✶✽

❚❤❡r❡ ❛r❡ t✇♦ r❡❛s♦♥s ✇❤② ❛ ♣r♦❞✉❝t ♦❢ ❍✉r❡✇✐❝③ s♣❛❝❡s X, Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❍✉r❡✇✐❝③✳ X × Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤✐s ♠❛② ✐♥❞❡❡❞ ❤❛♣♣❡♥✿ ✐♥ ❩❋❈ t❤❡r❡ ❛r❡ t✇♦ ♥♦r♠❛❧ s♣❛❝❡s X, Y ✇✐t❤ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ♠✉❝❤ str♦♥❣❡r t❤❛♥ t❤❡ ❍✉r❡✇✐❝③ ♦♥❡ s✉❝❤ t❤❛t X × Y ✐s ♥♦t ▲✐♥❞❡❧ö❢ ✭❚♦❞♦r❝❡✈✐❝ ✶✾✾✺✮✳ ✐s ▲✐♥❞❡❧ö❢✱ ❡✳❣✳✱ ✳ ❚❤✐s ❝❛s❡ ✐s s❡♥s✐t✐✈❡ t♦ t❤❡ ❛♠❜✐❡♥t s❡t✲t❤❡♦r❡t✐❝ ✉♥✐✈❡rs❡✿ ✉♥❞❡r ❈❍ t❤❡r❡ ❡①✐sts ❛ ❍✉r❡✇✐❝③ s♣❛❝❡ ✇❤♦s❡ sq✉❛r❡ ✐s ♥♦t ▼❡♥❣❡r✳ ❘❡q✉✐r❡s ❛s ♣r♦✈❡❞ ❜② ❙❝❤❡❡♣❡rs ❛♥❞ ❚s❛❜❛♥ ✐♥ ✷✵✵✷✳

❚❤❡♦r❡♠ ✭❘❡♣♦✈s✲❩✳ ✷✵✶✻✮

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❇♦r❡❧✬s ❝♦♥❥❡❝t✉r❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ ♠❡tr✐③❛❜❧❡ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt②✳ ❆s ❛ r❡s✉❧t✱ ✐♥ t❤✐s ♠♦❞❡❧ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt② ♣r♦✈✐❞❡❞ t❤❛t ✐t ✐s ▲✐♥❞❡❧ö❢ ◆♦t❡✿ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❇♦r❡❧✬s ❈♦♥❥❡❝t✉r❡✳

✶✺ ✴ ✶✽

❚❤❡r❡ ❛r❡ t✇♦ r❡❛s♦♥s ✇❤② ❛ ♣r♦❞✉❝t ♦❢ ❍✉r❡✇✐❝③ s♣❛❝❡s X, Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❍✉r❡✇✐❝③✳ X × Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤✐s ♠❛② ✐♥❞❡❡❞ ❤❛♣♣❡♥✿ ✐♥ ❩❋❈ t❤❡r❡ ❛r❡ t✇♦ ♥♦r♠❛❧ s♣❛❝❡s X, Y ✇✐t❤ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ♠✉❝❤ str♦♥❣❡r t❤❛♥ t❤❡ ❍✉r❡✇✐❝③ ♦♥❡ s✉❝❤ t❤❛t X × Y ✐s ♥♦t ▲✐♥❞❡❧ö❢ ✭❚♦❞♦r❝❡✈✐❝ ✶✾✾✺✮✳ X × Y ✐s ▲✐♥❞❡❧ö❢✱ ❡✳❣✳✱ X, Y ⊂ 2ω✳ ❚❤✐s ❝❛s❡ ✐s s❡♥s✐t✐✈❡ t♦ t❤❡ ❛♠❜✐❡♥t s❡t✲t❤❡♦r❡t✐❝ ✉♥✐✈❡rs❡✿ ✉♥❞❡r ❈❍ t❤❡r❡ ❡①✐sts ❛ ❍✉r❡✇✐❝③ s♣❛❝❡ ✇❤♦s❡ sq✉❛r❡ ✐s ♥♦t ▼❡♥❣❡r✳ ❘❡q✉✐r❡s ❛s ♣r♦✈❡❞ ❜② ❙❝❤❡❡♣❡rs ❛♥❞ ❚s❛❜❛♥ ✐♥ ✷✵✵✷✳

❚❤❡♦r❡♠ ✭❘❡♣♦✈s✲❩✳ ✷✵✶✻✮

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❇♦r❡❧✬s ❝♦♥❥❡❝t✉r❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ ♠❡tr✐③❛❜❧❡ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt②✳ ❆s ❛ r❡s✉❧t✱ ✐♥ t❤✐s ♠♦❞❡❧ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt② ♣r♦✈✐❞❡❞ t❤❛t ✐t ✐s ▲✐♥❞❡❧ö❢ ◆♦t❡✿ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❇♦r❡❧✬s ❈♦♥❥❡❝t✉r❡✳

✶✺ ✴ ✶✽

❚❤❡r❡ ❛r❡ t✇♦ r❡❛s♦♥s ✇❤② ❛ ♣r♦❞✉❝t ♦❢ ❍✉r❡✇✐❝③ s♣❛❝❡s X, Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❍✉r❡✇✐❝③✳ X × Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤✐s ♠❛② ✐♥❞❡❡❞ ❤❛♣♣❡♥✿ ✐♥ ❩❋❈ t❤❡r❡ ❛r❡ t✇♦ ♥♦r♠❛❧ s♣❛❝❡s X, Y ✇✐t❤ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ♠✉❝❤ str♦♥❣❡r t❤❛♥ t❤❡ ❍✉r❡✇✐❝③ ♦♥❡ s✉❝❤ t❤❛t X × Y ✐s ♥♦t ▲✐♥❞❡❧ö❢ ✭❚♦❞♦r❝❡✈✐❝ ✶✾✾✺✮✳ X × Y ✐s ▲✐♥❞❡❧ö❢✱ ❡✳❣✳✱ X, Y ⊂ 2ω✳ ❚❤✐s ❝❛s❡ ✐s s❡♥s✐t✐✈❡ t♦ t❤❡ ❛♠❜✐❡♥t s❡t✲t❤❡♦r❡t✐❝ ✉♥✐✈❡rs❡✿ ✉♥❞❡r ❈❍ t❤❡r❡ ❡①✐sts ❛ ❍✉r❡✇✐❝③ s♣❛❝❡ ✇❤♦s❡ sq✉❛r❡ ✐s ♥♦t ▼❡♥❣❡r✳ ❘❡q✉✐r❡s ❛s ♣r♦✈❡❞ ❜② ❙❝❤❡❡♣❡rs ❛♥❞ ❚s❛❜❛♥ ✐♥ ✷✵✵✷✳

❚❤❡♦r❡♠ ✭❘❡♣♦✈s✲❩✳ ✷✵✶✻✮

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❇♦r❡❧✬s ❝♦♥❥❡❝t✉r❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ ♠❡tr✐③❛❜❧❡ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt②✳ ❆s ❛ r❡s✉❧t✱ ✐♥ t❤✐s ♠♦❞❡❧ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt② ♣r♦✈✐❞❡❞ t❤❛t ✐t ✐s ▲✐♥❞❡❧ö❢ ◆♦t❡✿ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❇♦r❡❧✬s ❈♦♥❥❡❝t✉r❡✳

✶✺ ✴ ✶✽

❚❤❡r❡ ❛r❡ t✇♦ r❡❛s♦♥s ✇❤② ❛ ♣r♦❞✉❝t ♦❢ ❍✉r❡✇✐❝③ s♣❛❝❡s X, Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❍✉r❡✇✐❝③✳ X × Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤✐s ♠❛② ✐♥❞❡❡❞ ❤❛♣♣❡♥✿ ✐♥ ❩❋❈ t❤❡r❡ ❛r❡ t✇♦ ♥♦r♠❛❧ s♣❛❝❡s X, Y ✇✐t❤ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ♠✉❝❤ str♦♥❣❡r t❤❛♥ t❤❡ ❍✉r❡✇✐❝③ ♦♥❡ s✉❝❤ t❤❛t X × Y ✐s ♥♦t ▲✐♥❞❡❧ö❢ ✭❚♦❞♦r❝❡✈✐❝ ✶✾✾✺✮✳ X × Y ✐s ▲✐♥❞❡❧ö❢✱ ❡✳❣✳✱ X, Y ⊂ 2ω✳ ❚❤✐s ❝❛s❡ ✐s s❡♥s✐t✐✈❡ t♦ t❤❡ ❛♠❜✐❡♥t s❡t✲t❤❡♦r❡t✐❝ ✉♥✐✈❡rs❡✿ ✉♥❞❡r ❈❍ t❤❡r❡ ❡①✐sts ❛ ❍✉r❡✇✐❝③ s♣❛❝❡ ✇❤♦s❡ sq✉❛r❡ ✐s ♥♦t ▼❡♥❣❡r✳ ❘❡q✉✐r❡s ❛s ♣r♦✈❡❞ ❜② ❙❝❤❡❡♣❡rs ❛♥❞ ❚s❛❜❛♥ ✐♥ ✷✵✵✷✳

❚❤❡♦r❡♠ ✭❘❡♣♦✈s✲❩✳ ✷✵✶✻✮

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❇♦r❡❧✬s ❝♦♥❥❡❝t✉r❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ ♠❡tr✐③❛❜❧❡ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt②✳ ❆s ❛ r❡s✉❧t✱ ✐♥ t❤✐s ♠♦❞❡❧ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt② ♣r♦✈✐❞❡❞ t❤❛t ✐t ✐s ▲✐♥❞❡❧ö❢ ◆♦t❡✿ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❇♦r❡❧✬s ❈♦♥❥❡❝t✉r❡✳

✶✺ ✴ ✶✽

❚❤❡r❡ ❛r❡ t✇♦ r❡❛s♦♥s ✇❤② ❛ ♣r♦❞✉❝t ♦❢ ❍✉r❡✇✐❝③ s♣❛❝❡s X, Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❍✉r❡✇✐❝③✳ X × Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤✐s ♠❛② ✐♥❞❡❡❞ ❤❛♣♣❡♥✿ ✐♥ ❩❋❈ t❤❡r❡ ❛r❡ t✇♦ ♥♦r♠❛❧ s♣❛❝❡s X, Y ✇✐t❤ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ♠✉❝❤ str♦♥❣❡r t❤❛♥ t❤❡ ❍✉r❡✇✐❝③ ♦♥❡ s✉❝❤ t❤❛t X × Y ✐s ♥♦t ▲✐♥❞❡❧ö❢ ✭❚♦❞♦r❝❡✈✐❝ ✶✾✾✺✮✳ X × Y ✐s ▲✐♥❞❡❧ö❢✱ ❡✳❣✳✱ X, Y ⊂ 2ω✳ ❚❤✐s ❝❛s❡ ✐s s❡♥s✐t✐✈❡ t♦ t❤❡ ❛♠❜✐❡♥t s❡t✲t❤❡♦r❡t✐❝ ✉♥✐✈❡rs❡✿ ✉♥❞❡r ❈❍ t❤❡r❡ ❡①✐sts ❛ ❍✉r❡✇✐❝③ s♣❛❝❡ ✇❤♦s❡ sq✉❛r❡ ✐s ♥♦t ▼❡♥❣❡r✳ ❘❡q✉✐r❡s cov(N) = cof(N) ❛s ♣r♦✈❡❞ ❜② ❙❝❤❡❡♣❡rs ❛♥❞ ❚s❛❜❛♥ ✐♥ ✷✵✵✷✳

❚❤❡♦r❡♠ ✭❘❡♣♦✈s✲❩✳ ✷✵✶✻✮

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❇♦r❡❧✬s ❝♦♥❥❡❝t✉r❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ ♠❡tr✐③❛❜❧❡ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt②✳ ❆s ❛ r❡s✉❧t✱ ✐♥ t❤✐s ♠♦❞❡❧ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt② ♣r♦✈✐❞❡❞ t❤❛t ✐t ✐s ▲✐♥❞❡❧ö❢ ◆♦t❡✿ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❇♦r❡❧✬s ❈♦♥❥❡❝t✉r❡✳

✶✺ ✴ ✶✽

❚❤❡r❡ ❛r❡ t✇♦ r❡❛s♦♥s ✇❤② ❛ ♣r♦❞✉❝t ♦❢ ❍✉r❡✇✐❝③ s♣❛❝❡s X, Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❍✉r❡✇✐❝③✳ X × Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤✐s ♠❛② ✐♥❞❡❡❞ ❤❛♣♣❡♥✿ ✐♥ ❩❋❈ t❤❡r❡ ❛r❡ t✇♦ ♥♦r♠❛❧ s♣❛❝❡s X, Y ✇✐t❤ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ♠✉❝❤ str♦♥❣❡r t❤❛♥ t❤❡ ❍✉r❡✇✐❝③ ♦♥❡ s✉❝❤ t❤❛t X × Y ✐s ♥♦t ▲✐♥❞❡❧ö❢ ✭❚♦❞♦r❝❡✈✐❝ ✶✾✾✺✮✳ X × Y ✐s ▲✐♥❞❡❧ö❢✱ ❡✳❣✳✱ X, Y ⊂ 2ω✳ ❚❤✐s ❝❛s❡ ✐s s❡♥s✐t✐✈❡ t♦ t❤❡ ❛♠❜✐❡♥t s❡t✲t❤❡♦r❡t✐❝ ✉♥✐✈❡rs❡✿ ✉♥❞❡r ❈❍ t❤❡r❡ ❡①✐sts ❛ ❍✉r❡✇✐❝③ s♣❛❝❡ ✇❤♦s❡ sq✉❛r❡ ✐s ♥♦t ▼❡♥❣❡r✳ ❘❡q✉✐r❡s cov(N) = cof(N) ❛s ♣r♦✈❡❞ ❜② ❙❝❤❡❡♣❡rs ❛♥❞ ❚s❛❜❛♥ ✐♥ ✷✵✵✷✳

❚❤❡♦r❡♠ ✭❘❡♣♦✈s✲❩✳ ✷✵✶✻✮

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❇♦r❡❧✬s ❝♦♥❥❡❝t✉r❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ ♠❡tr✐③❛❜❧❡ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt②✳ ❆s ❛ r❡s✉❧t✱ ✐♥ t❤✐s ♠♦❞❡❧ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt② ♣r♦✈✐❞❡❞ t❤❛t ✐t ✐s ▲✐♥❞❡❧ö❢ ◆♦t❡✿ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❇♦r❡❧✬s ❈♦♥❥❡❝t✉r❡✳

✶✺ ✴ ✶✽

❚❤❡r❡ ❛r❡ t✇♦ r❡❛s♦♥s ✇❤② ❛ ♣r♦❞✉❝t ♦❢ ❍✉r❡✇✐❝③ s♣❛❝❡s X, Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❍✉r❡✇✐❝③✳ X × Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤✐s ♠❛② ✐♥❞❡❡❞ ❤❛♣♣❡♥✿ ✐♥ ❩❋❈ t❤❡r❡ ❛r❡ t✇♦ ♥♦r♠❛❧ s♣❛❝❡s X, Y ✇✐t❤ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ♠✉❝❤ str♦♥❣❡r t❤❛♥ t❤❡ ❍✉r❡✇✐❝③ ♦♥❡ s✉❝❤ t❤❛t X × Y ✐s ♥♦t ▲✐♥❞❡❧ö❢ ✭❚♦❞♦r❝❡✈✐❝ ✶✾✾✺✮✳ X × Y ✐s ▲✐♥❞❡❧ö❢✱ ❡✳❣✳✱ X, Y ⊂ 2ω✳ ❚❤✐s ❝❛s❡ ✐s s❡♥s✐t✐✈❡ t♦ t❤❡ ❛♠❜✐❡♥t s❡t✲t❤❡♦r❡t✐❝ ✉♥✐✈❡rs❡✿ ✉♥❞❡r ❈❍ t❤❡r❡ ❡①✐sts ❛ ❍✉r❡✇✐❝③ s♣❛❝❡ ✇❤♦s❡ sq✉❛r❡ ✐s ♥♦t ▼❡♥❣❡r✳ ❘❡q✉✐r❡s cov(N) = cof(N) ❛s ♣r♦✈❡❞ ❜② ❙❝❤❡❡♣❡rs ❛♥❞ ❚s❛❜❛♥ ✐♥ ✷✵✵✷✳

❚❤❡♦r❡♠ ✭❘❡♣♦✈s✲❩✳ ✷✵✶✻✮

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❇♦r❡❧✬s ❝♦♥❥❡❝t✉r❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ ♠❡tr✐③❛❜❧❡ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt②✳ ❆s ❛ r❡s✉❧t✱ ✐♥ t❤✐s ♠♦❞❡❧ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt② ♣r♦✈✐❞❡❞ t❤❛t ✐t ✐s ▲✐♥❞❡❧ö❢ ◆♦t❡✿ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❇♦r❡❧✬s ❈♦♥❥❡❝t✉r❡✳

✶✺ ✴ ✶✽

❚❤❡r❡ ❛r❡ t✇♦ r❡❛s♦♥s ✇❤② ❛ ♣r♦❞✉❝t ♦❢ ❍✉r❡✇✐❝③ s♣❛❝❡s X, Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❍✉r❡✇✐❝③✳ X × Y ♠❛② ❢❛✐❧ t♦ ❜❡ ❛ ▲✐♥❞❡❧ö❢ s♣❛❝❡✳ ❚❤✐s ♠❛② ✐♥❞❡❡❞ ❤❛♣♣❡♥✿ ✐♥ ❩❋❈ t❤❡r❡ ❛r❡ t✇♦ ♥♦r♠❛❧ s♣❛❝❡s X, Y ✇✐t❤ ❛ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ♠✉❝❤ str♦♥❣❡r t❤❛♥ t❤❡ ❍✉r❡✇✐❝③ ♦♥❡ s✉❝❤ t❤❛t X × Y ✐s ♥♦t ▲✐♥❞❡❧ö❢ ✭❚♦❞♦r❝❡✈✐❝ ✶✾✾✺✮✳ X × Y ✐s ▲✐♥❞❡❧ö❢✱ ❡✳❣✳✱ X, Y ⊂ 2ω✳ ❚❤✐s ❝❛s❡ ✐s s❡♥s✐t✐✈❡ t♦ t❤❡ ❛♠❜✐❡♥t s❡t✲t❤❡♦r❡t✐❝ ✉♥✐✈❡rs❡✿ ✉♥❞❡r ❈❍ t❤❡r❡ ❡①✐sts ❛ ❍✉r❡✇✐❝③ s♣❛❝❡ ✇❤♦s❡ sq✉❛r❡ ✐s ♥♦t ▼❡♥❣❡r✳ ❘❡q✉✐r❡s cov(N) = cof(N) ❛s ♣r♦✈❡❞ ❜② ❙❝❤❡❡♣❡rs ❛♥❞ ❚s❛❜❛♥ ✐♥ ✷✵✵✷✳

❚❤❡♦r❡♠ ✭❘❡♣♦✈s✲❩✳ ✷✵✶✻✮

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❇♦r❡❧✬s ❝♦♥❥❡❝t✉r❡✱ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ ♠❡tr✐③❛❜❧❡ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt②✳ ❆s ❛ r❡s✉❧t✱ ✐♥ t❤✐s ♠♦❞❡❧ t❤❡ ♣r♦❞✉❝t ♦❢ ❛♥② t✇♦ ❍✉r❡✇✐❝③ s♣❛❝❡s ❤❛s t❤❡ ▼❡♥❣❡r ♣r♦♣❡rt② ♣r♦✈✐❞❡❞ t❤❛t ✐t ✐s ▲✐♥❞❡❧ö❢ ◆♦t❡✿ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❞♦❡s♥✬t ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❇♦r❡❧✬s ❈♦♥❥❡❝t✉r❡✳

✶✺ ✴ ✶✽

❲❡❛❦ ❝♦♥❝❡♥tr❛t✐♦♥

❉❡✜♥✐t✐♦♥

X ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ✐❢ ❢♦r ❡✈❡r② ❝♦❧❧❡❝t✐♦♥ Q ⊂ [X]ω ✇❤✐❝❤ ✐s ❝♦✜♥❛❧ ✇✐t❤ r❡s♣❡❝t t♦ ✐♥❝❧✉s✐♦♥✱ ❛♥❞ ❢♦r ❡✈❡r② ❢✉♥❝t✐♦♥ ❛ss✐❣♥✐♥❣ t♦ ❡❛❝❤ ❛ ✲s❡t ❝♦♥t❛✐♥✐♥❣ ✱ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t ✳ ❯♥❞❡r ❈❍ ❛♥② s✉❜s❡t ♦❢ ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ❙♦ t❤❡ ♥♦t✐♦♥ ♠✐❣❤t ❜❡ ✐♥t❡r❡st✐♥❣ ♦♥❧② ✉♥❞❡r ✳

▲❡♠♠❛

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❡✈❡r② ❍✉r❡✇✐❝③ s✉❜s♣❛❝❡ ♦❢ ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ■❢ ✱ t❤❡♥ ❛ ♣r♦❞✉❝t ♦❢ ❛ ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ❛♥❞ ❛ ❍✉r❡✇✐❝③ ✐s ▼❡♥❣❡r✳

✶✻ ✴ ✶✽

❲❡❛❦ ❝♦♥❝❡♥tr❛t✐♦♥

❉❡✜♥✐t✐♦♥

X ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ✐❢ ❢♦r ❡✈❡r② ❝♦❧❧❡❝t✐♦♥ Q ⊂ [X]ω ✇❤✐❝❤ ✐s ❝♦✜♥❛❧ ✇✐t❤ r❡s♣❡❝t t♦ ✐♥❝❧✉s✐♦♥✱ ❛♥❞ ❢♦r ❡✈❡r② ❢✉♥❝t✐♦♥ R : Q → P(X) ❛ss✐❣♥✐♥❣ t♦ ❡❛❝❤ Q ∈ Q ❛ Gδ✲s❡t R(Q) ❝♦♥t❛✐♥✐♥❣ Q✱ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t ✳ ❯♥❞❡r ❈❍ ❛♥② s✉❜s❡t ♦❢ ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ❙♦ t❤❡ ♥♦t✐♦♥ ♠✐❣❤t ❜❡ ✐♥t❡r❡st✐♥❣ ♦♥❧② ✉♥❞❡r ✳

▲❡♠♠❛

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❡✈❡r② ❍✉r❡✇✐❝③ s✉❜s♣❛❝❡ ♦❢ ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ■❢ ✱ t❤❡♥ ❛ ♣r♦❞✉❝t ♦❢ ❛ ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ❛♥❞ ❛ ❍✉r❡✇✐❝③ ✐s ▼❡♥❣❡r✳

✶✻ ✴ ✶✽

❲❡❛❦ ❝♦♥❝❡♥tr❛t✐♦♥

❉❡✜♥✐t✐♦♥

X ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ✐❢ ❢♦r ❡✈❡r② ❝♦❧❧❡❝t✐♦♥ Q ⊂ [X]ω ✇❤✐❝❤ ✐s ❝♦✜♥❛❧ ✇✐t❤ r❡s♣❡❝t t♦ ✐♥❝❧✉s✐♦♥✱ ❛♥❞ ❢♦r ❡✈❡r② ❢✉♥❝t✐♦♥ R : Q → P(X) ❛ss✐❣♥✐♥❣ t♦ ❡❛❝❤ Q ∈ Q ❛ Gδ✲s❡t R(Q) ❝♦♥t❛✐♥✐♥❣ Q✱ t❤❡r❡ ❡①✐sts Q1 ∈ [Q]ω1 s✉❝❤ t❤❛t X ⊂

Q∈Q1 R(Q)✳

❯♥❞❡r ❈❍ ❛♥② s✉❜s❡t ♦❢ ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ❙♦ t❤❡ ♥♦t✐♦♥ ♠✐❣❤t ❜❡ ✐♥t❡r❡st✐♥❣ ♦♥❧② ✉♥❞❡r ✳

▲❡♠♠❛

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❡✈❡r② ❍✉r❡✇✐❝③ s✉❜s♣❛❝❡ ♦❢ ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ■❢ ✱ t❤❡♥ ❛ ♣r♦❞✉❝t ♦❢ ❛ ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ❛♥❞ ❛ ❍✉r❡✇✐❝③ ✐s ▼❡♥❣❡r✳

✶✻ ✴ ✶✽

❲❡❛❦ ❝♦♥❝❡♥tr❛t✐♦♥

❉❡✜♥✐t✐♦♥

X ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ✐❢ ❢♦r ❡✈❡r② ❝♦❧❧❡❝t✐♦♥ Q ⊂ [X]ω ✇❤✐❝❤ ✐s ❝♦✜♥❛❧ ✇✐t❤ r❡s♣❡❝t t♦ ✐♥❝❧✉s✐♦♥✱ ❛♥❞ ❢♦r ❡✈❡r② ❢✉♥❝t✐♦♥ R : Q → P(X) ❛ss✐❣♥✐♥❣ t♦ ❡❛❝❤ Q ∈ Q ❛ Gδ✲s❡t R(Q) ❝♦♥t❛✐♥✐♥❣ Q✱ t❤❡r❡ ❡①✐sts Q1 ∈ [Q]ω1 s✉❝❤ t❤❛t X ⊂

Q∈Q1 R(Q)✳

❯♥❞❡r ❈❍ ❛♥② s✉❜s❡t ♦❢ 2ω ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ❙♦ t❤❡ ♥♦t✐♦♥ ♠✐❣❤t ❜❡ ✐♥t❡r❡st✐♥❣ ♦♥❧② ✉♥❞❡r ✳

▲❡♠♠❛

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❡✈❡r② ❍✉r❡✇✐❝③ s✉❜s♣❛❝❡ ♦❢ ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ■❢ ✱ t❤❡♥ ❛ ♣r♦❞✉❝t ♦❢ ❛ ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ❛♥❞ ❛ ❍✉r❡✇✐❝③ ✐s ▼❡♥❣❡r✳

✶✻ ✴ ✶✽

❲❡❛❦ ❝♦♥❝❡♥tr❛t✐♦♥

❉❡✜♥✐t✐♦♥

X ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ✐❢ ❢♦r ❡✈❡r② ❝♦❧❧❡❝t✐♦♥ Q ⊂ [X]ω ✇❤✐❝❤ ✐s ❝♦✜♥❛❧ ✇✐t❤ r❡s♣❡❝t t♦ ✐♥❝❧✉s✐♦♥✱ ❛♥❞ ❢♦r ❡✈❡r② ❢✉♥❝t✐♦♥ R : Q → P(X) ❛ss✐❣♥✐♥❣ t♦ ❡❛❝❤ Q ∈ Q ❛ Gδ✲s❡t R(Q) ❝♦♥t❛✐♥✐♥❣ Q✱ t❤❡r❡ ❡①✐sts Q1 ∈ [Q]ω1 s✉❝❤ t❤❛t X ⊂

Q∈Q1 R(Q)✳

❯♥❞❡r ❈❍ ❛♥② s✉❜s❡t ♦❢ 2ω ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ❙♦ t❤❡ ♥♦t✐♦♥ ♠✐❣❤t ❜❡ ✐♥t❡r❡st✐♥❣ ♦♥❧② ✉♥❞❡r c > ω1✳

▲❡♠♠❛

■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❡✈❡r② ❍✉r❡✇✐❝③ s✉❜s♣❛❝❡ ♦❢ ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ■❢ ✱ t❤❡♥ ❛ ♣r♦❞✉❝t ♦❢ ❛ ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ❛♥❞ ❛ ❍✉r❡✇✐❝③ ✐s ▼❡♥❣❡r✳

✶✻ ✴ ✶✽

❲❡❛❦ ❝♦♥❝❡♥tr❛t✐♦♥

❉❡✜♥✐t✐♦♥

X ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ✐❢ ❢♦r ❡✈❡r② ❝♦❧❧❡❝t✐♦♥ Q ⊂ [X]ω ✇❤✐❝❤ ✐s ❝♦✜♥❛❧ ✇✐t❤ r❡s♣❡❝t t♦ ✐♥❝❧✉s✐♦♥✱ ❛♥❞ ❢♦r ❡✈❡r② ❢✉♥❝t✐♦♥ R : Q → P(X) ❛ss✐❣♥✐♥❣ t♦ ❡❛❝❤ Q ∈ Q ❛ Gδ✲s❡t R(Q) ❝♦♥t❛✐♥✐♥❣ Q✱ t❤❡r❡ ❡①✐sts Q1 ∈ [Q]ω1 s✉❝❤ t❤❛t X ⊂

Q∈Q1 R(Q)✳

❯♥❞❡r ❈❍ ❛♥② s✉❜s❡t ♦❢ 2ω ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ❙♦ t❤❡ ♥♦t✐♦♥ ♠✐❣❤t ❜❡ ✐♥t❡r❡st✐♥❣ ♦♥❧② ✉♥❞❡r c > ω1✳

▲❡♠♠❛

◮ ■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❡✈❡r② ❍✉r❡✇✐❝③ s✉❜s♣❛❝❡ ♦❢ P(ω) ✐s ✇❡❛❦❧②

❝♦♥❝❡♥tr❛t❡❞✳ ■❢ ✱ t❤❡♥ ❛ ♣r♦❞✉❝t ♦❢ ❛ ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ❛♥❞ ❛ ❍✉r❡✇✐❝③ ✐s ▼❡♥❣❡r✳

✶✻ ✴ ✶✽

❲❡❛❦ ❝♦♥❝❡♥tr❛t✐♦♥

❉❡✜♥✐t✐♦♥

X ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ ✐❢ ❢♦r ❡✈❡r② ❝♦❧❧❡❝t✐♦♥ Q ⊂ [X]ω ✇❤✐❝❤ ✐s ❝♦✜♥❛❧ ✇✐t❤ r❡s♣❡❝t t♦ ✐♥❝❧✉s✐♦♥✱ ❛♥❞ ❢♦r ❡✈❡r② ❢✉♥❝t✐♦♥ R : Q → P(X) ❛ss✐❣♥✐♥❣ t♦ ❡❛❝❤ Q ∈ Q ❛ Gδ✲s❡t R(Q) ❝♦♥t❛✐♥✐♥❣ Q✱ t❤❡r❡ ❡①✐sts Q1 ∈ [Q]ω1 s✉❝❤ t❤❛t X ⊂

Q∈Q1 R(Q)✳

❯♥❞❡r ❈❍ ❛♥② s✉❜s❡t ♦❢ 2ω ✐s ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞✳ ❙♦ t❤❡ ♥♦t✐♦♥ ♠✐❣❤t ❜❡ ✐♥t❡r❡st✐♥❣ ♦♥❧② ✉♥❞❡r c > ω1✳

▲❡♠♠❛

◮ ■♥ t❤❡ ▲❛✈❡r ♠♦❞❡❧ ❡✈❡r② ❍✉r❡✇✐❝③ s✉❜s♣❛❝❡ ♦❢ P(ω) ✐s ✇❡❛❦❧②

❝♦♥❝❡♥tr❛t❡❞✳

◮ ■❢ b > ω1✱ t❤❡♥ ❛ ♣r♦❞✉❝t ♦❢ ❛ ✇❡❛❦❧② ❝♦♥❝❡♥tr❛t❡❞ X ⊂ 2ω

❛♥❞ ❛ ❍✉r❡✇✐❝③ Y ⊂ 2ω ✐s ▼❡♥❣❡r✳

✶✻ ✴ ✶✽

❍♦✇ ❝♦♥❝❡♥tr❛t✐♦♥ ✇♦r❦s ✐♥ ♣r♦❞✉❝ts

❚✐♠❡ ♣❡r♠✐tt✐♥❣✱ ✐t s❤♦✉❧❞ ❜❡ ❡①♣❧❛✐♥❡❞ ♦♥ t❤❡ ❜❧❛❝❦❜♦❛r❞ ✇❤② ❍✉r❡✇✐❝③ ① ❝♦♥❝❡♥tr❛t❡❞ ✐s ▼❡♥❣❡r✳

✶✼ ✴ ✶✽

❚❤❡ ❧❛st s❧✐❞❡

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳

✶✽ ✴ ✶✽