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

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

❆ 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 ∈ ω ω ✱ 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 → ω ω ✳ ✸ ✴ ✶✽

✐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 ∈ ω ω ✱ 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 → ω ω ✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f [ X ] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ≤ ∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ω ω ✳ ✸ ✴ ✶✽

✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ s✉❜s❡t ♦❢ ✳ ✐s ❍✉r❡✇✐❝③✳ ✐s ▼❡♥❣❡r✳ ❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ●✐✈❡♥ x, y ∈ ω ω ✱ 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 → ω ω ✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f [ X ] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ≤ ∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ω ω ✳ b ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ω ω ✳ ✸ ✴ ✶✽

✐s ❍✉r❡✇✐❝③✳ ✐s ▼❡♥❣❡r✳ ❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ●✐✈❡♥ x, y ∈ ω ω ✱ 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 → ω ω ✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f [ X ] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ≤ ∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ω ω ✳ b ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ω ω ✳ d ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ s✉❜s❡t ♦❢ ω ω ✳ ✸ ✴ ✶✽

✐s ▼❡♥❣❡r✳ ❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ●✐✈❡♥ x, y ∈ ω ω ✱ 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 → ω ω ✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f [ X ] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ≤ ∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ω ω ✳ b ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ω ω ✳ d ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ s✉❜s❡t ♦❢ ω ω ✳ | X | < b → X ✐s ❍✉r❡✇✐❝③✳ ✸ ✴ ✶✽

❆ ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ●✐✈❡♥ x, y ∈ ω ω ✱ 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 → ω ω ✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f [ X ] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ≤ ∗ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ω ω ✳ b ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ω ω ✳ d ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ s✉❜s❡t ♦❢ ω ω ✳ | X | < b → X ✐s ❍✉r❡✇✐❝③✳ | X | < d → X ✐s ▼❡♥❣❡r✳ ✸ ✴ ✶✽

❊✈❡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 ⊂ ω ω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ M | ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M ✳ ✹ ✴ ✶✽

✐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 ⊂ ω ω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ M | ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M ✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛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 ⊂ ω ω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ M | ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M ✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ X ⊂ 2 ω ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ N | ≤ ω ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t N ✳ ✹ ✴ ✶✽

❈♦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 ⊂ ω ω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ M | ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M ✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ X ⊂ 2 ω ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ N | ≤ ω ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t N ✳ ❚❤❡♦r❡♠ ✭❙❝❤❡❡♣❡rs ✶✾✾✻✮ ▲❡t P ❜❡ ❝♦♠♣❛❝t✳ X ⊂ P ✐s ❍✉r❡✇✐❝③ ✐✛ ❢♦r ❡✈❡r② G δ ✲s❡t G ⊃ X t❤❡r❡ ❡①✐sts ❛ σ ✲❝♦♠♣❛❝t F s✉❝❤ t❤❛t X ⊂ F ⊂ G ✳ ✹ ✴ ✶✽

❙✐❡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 ⊂ ω ω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ M | ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M ✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ X ⊂ 2 ω ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ N | ≤ ω ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t N ✳ ❚❤❡♦r❡♠ ✭❙❝❤❡❡♣❡rs ✶✾✾✻✮ ▲❡t P ❜❡ ❝♦♠♣❛❝t✳ X ⊂ P ✐s ❍✉r❡✇✐❝③ ✐✛ ❢♦r ❡✈❡r② G δ ✲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❡ ❣❡♥❡r❛❧❧②✿ ✲❙✐❡r♣✐♥s❦✐ s❡ts ❛r❡ ❍✉r❡✇✐❝③ ❛♥❞ ✲▲✉③✐♥ s❡ts ❛r❡ ▼❡♥❣❡r✳ ❊①❛♠♣❧❡s ✉♥❞❡r ❈❍✳ X ⊂ ω ω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ M | ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M ✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ X ⊂ 2 ω ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ N | ≤ ω ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t N ✳ ❚❤❡♦r❡♠ ✭❙❝❤❡❡♣❡rs ✶✾✾✻✮ ▲❡t P ❜❡ ❝♦♠♣❛❝t✳ X ⊂ P ✐s ❍✉r❡✇✐❝③ ✐✛ ❢♦r ❡✈❡r② G δ ✲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✳ ✹ ✴ ✶✽

❊①❛♠♣❧❡s ✉♥❞❡r ❈❍✳ X ⊂ ω ω ✐s ❛ ▲✉③✐♥ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ M | ≤ ω ❢♦r ❛♥② ♠❡❛❣❡r M ✳ ❊✈❡r② ▲✉③✐♥ s❡t ✐s ▼❡♥❣❡r ❜❡❝❛✉s❡ ❝♦♥❝❡♥tr❛t❡❞✳ X ⊂ 2 ω ✐s ❛ ❙✐❡r♣✐♥s❦✐ s❡t ✐❢ | X | = ω 1 ❛♥❞ | X ∩ N | ≤ ω ❢♦r ❛♥② ♠❡❛s✉r❡ ✵ s❡t N ✳ ❚❤❡♦r❡♠ ✭❙❝❤❡❡♣❡rs ✶✾✾✻✮ ▲❡t P ❜❡ ❝♦♠♣❛❝t✳ X ⊂ P ✐s ❍✉r❡✇✐❝③ ✐✛ ❢♦r ❡✈❡r② G δ ✲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✳ ✹ ✴ ✶✽

■❢ ✱ 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 ⊂ ω ω ✐s κ ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q ✱ ✐❢ | X | ≥ κ ❛♥❞ | X \ U | < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ω ω ❝♦♥t❛✐♥✐♥❣ Q ✳ ✺ ✴ ✶✽

❋❛❝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 ⊂ ω ω ✐s κ ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q ✱ ✐❢ | X | ≥ κ ❛♥❞ | X \ U | < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ω ω ❝♦♥t❛✐♥✐♥❣ Q ✳ ■❢ κ ≤ d ✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡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 ⊂ ω ω ✐s κ ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q ✱ ✐❢ | X | ≥ κ ❛♥❞ | X \ U | < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ω ω ❝♦♥t❛✐♥✐♥❣ Q ✳ ■❢ κ ≤ d ✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ { d α : α < d } ⊂ ω ω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = { s α : α < d } ⊂ ω ↑ ω s✉❝❤ t❤❛t s α �≤ ∗ d β ❢♦r ❛❧❧ β ≤ α ✳ ✺ ✴ ✶✽

❋❛❝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 ⊂ ω ω ✐s κ ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q ✱ ✐❢ | X | ≥ κ ❛♥❞ | X \ U | < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ω ω ❝♦♥t❛✐♥✐♥❣ Q ✳ ■❢ κ ≤ d ✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ { d α : α < d } ⊂ ω ω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = { s α : α < d } ⊂ ω ↑ ω s✉❝❤ t❤❛t s α �≤ ∗ d β ❢♦r ❛❧❧ β ≤ α ✳ ❱✐❡✇❡❞ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ ( ω + 1) ↑ ω ✱ S ✐s d ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q = { x ∈ ( ω + 1) ↑ ω : x ✐s ❡✈❡♥t✉❛❧❧② ω } ✳ ✷ ✺ ✴ ✶✽

✐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 ⊂ ω ω ✐s κ ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q ✱ ✐❢ | X | ≥ κ ❛♥❞ | X \ U | < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ω ω ❝♦♥t❛✐♥✐♥❣ Q ✳ ■❢ κ ≤ d ✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ { d α : α < d } ⊂ ω ω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = { s α : α < d } ⊂ ω ↑ ω s✉❝❤ t❤❛t s α �≤ ∗ d β ❢♦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 = { b α : α < b } ⊂ ω ω s✉❝❤ t❤❛t b β ≤ ∗ b α ❢♦r ❛❧❧ β ≤ α ✳ ✺ ✴ ✶✽

◆♦♥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 ⊂ ω ω ✐s κ ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q ✱ ✐❢ | X | ≥ κ ❛♥❞ | X \ U | < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ω ω ❝♦♥t❛✐♥✐♥❣ Q ✳ ■❢ κ ≤ d ✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ { d α : α < d } ⊂ ω ω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = { s α : α < d } ⊂ ω ↑ ω s✉❝❤ t❤❛t s α �≤ ∗ d β ❢♦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 = { b α : α < b } ⊂ ω ω s✉❝❤ t❤❛t b β ≤ ∗ b α ❢♦r ❛❧❧ β ≤ α ✳ B ✐s b ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ Q ✳ ✷ ✺ ✴ ✶✽

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

✇r♦♥❣ ✉♥❞❡r ❈❍✱ tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳ ❩❋❈ ❡①❛♠♣❧❡s ❆ s❡t X ⊂ ω ω ✐s κ ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q ✱ ✐❢ | X | ≥ κ ❛♥❞ | X \ U | < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ω ω ❝♦♥t❛✐♥✐♥❣ Q ✳ ■❢ κ ≤ d ✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ { d α : α < d } ⊂ ω ω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = { s α : α < d } ⊂ ω ↑ ω s✉❝❤ t❤❛t s α �≤ ∗ d β ❢♦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 = { b α : α < b } ⊂ ω ω s✉❝❤ t❤❛t b β ≤ ∗ b α ❢♦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✿ ✺ ✴ ✶✽

tr✉❡ ✐♥ t❤❡ ▼✐❧❧❡r ♠♦❞❡❧✳ ❩❋❈ ❡①❛♠♣❧❡s ❆ s❡t X ⊂ ω ω ✐s κ ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q ✱ ✐❢ | X | ≥ κ ❛♥❞ | X \ U | < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ω ω ❝♦♥t❛✐♥✐♥❣ Q ✳ ■❢ κ ≤ d ✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ { d α : α < d } ⊂ ω ω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = { s α : α < d } ⊂ ω ↑ ω s✉❝❤ t❤❛t s α �≤ ∗ d β ❢♦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 = { b α : α < b } ⊂ ω ω s✉❝❤ t❤❛t b β ≤ ∗ b α ❢♦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 ❈❍✱ ✺ ✴ ✶✽

❩❋❈ ❡①❛♠♣❧❡s ❆ s❡t X ⊂ ω ω ✐s κ ✲❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❛ ❝♦✉♥t❛❜❧❡ Q ✱ ✐❢ | X | ≥ κ ❛♥❞ | X \ U | < κ ❢♦r ❛♥② ♦♣❡♥ U ⊂ ω ω ❝♦♥t❛✐♥✐♥❣ Q ✳ ■❢ κ ≤ d ✱ t❤❡♥ X ∪ Q ✐s ▼❡♥❣❡r✳ ❋❛❝t✳ ❚❤❡r❡ ❡①✐sts ❛ d ✲❝♦♥❝❡♥tr❛t❡ s❡t✳ Pr♦♦❢✳ ❋✐① ❛ ❞♦♠✐♥❛t✐♥❣ { d α : α < d } ⊂ ω ω ❛♥❞ ✐♥❞✉❝t✐✈❡❧② ❝♦♥str✉❝t S = { s α : α < d } ⊂ ω ↑ ω s✉❝❤ t❤❛t s α �≤ ∗ d β ❢♦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 = { b α : α < b } ⊂ ω ω s✉❝❤ t❤❛t b β ≤ ∗ b α ❢♦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 ♠♦❞❡❧✳ ✺ ✴ ✶✽

✐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✱ ✻ ✴ ✶✽

❛♥❞ ✉♥❞❡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✱ ✻ ✴ ✶✽

❝♦♥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✳ ✻ ✴ ✶✽

❆ ❝♦♥❞✐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✳ M F ❝♦♥s✐sts ♦❢ ♣❛✐rs � s, F � s✉❝❤ t❤❛t s ∈ [ ω ] <ω ✱ F ∈ F ✱ ❛♥❞ max s < min F ✳ ✻ ✴ ✶✽

✳ ✐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✳ M F ❝♦♥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✳ M F ❝♦♥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 ❛ ♥❛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✳ M F ❝♦♥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 ✳ M F ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐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✳ M F ❝♦♥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 ✳ M F ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F ✳ M F ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ 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✳ M F ❝♦♥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 ✳ M F ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F ✳ M F ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ F ✿ ✐❢ G ✐s ❛ M F ✲❣❡♥❡r✐❝✱ t❤❡♥ X = � { s : ∃ F ∈ F ( � s, F � ∈ G ) } ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② F ∈ F ✳ ✻ ✴ ✶✽

❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡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✳ M F ❝♦♥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 ✳ M F ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F ✳ M F ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ F ✿ ✐❢ G ✐s ❛ M F ✲❣❡♥❡r✐❝✱ t❤❡♥ X = � { s : ∃ F ∈ F ( � s, F � ∈ G ) } ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② F ∈ F ✳ ❆♣♣❧✐❝❛t✐♦♥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✳ M F ❝♦♥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 ✳ M F ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F ✳ M F ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ F ✿ ✐❢ G ✐s ❛ M F ✲❣❡♥❡r✐❝✱ t❤❡♥ X = � { s : ∃ F ∈ F ( � s, F � ∈ G ) } ✐s ❛❧♠♦st ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② F ∈ F ✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ❦✐❧❧✐♥❣ ♠❛❞ ❢❛♠✐❧✐❡s✱ ✻ ✴ ✶✽

▼❛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✳ M F ❝♦♥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 ✳ M F ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ ▼❛t❤✐❛s ❢♦r❝✐♥❣ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F ✳ M F ✐s ❛ ♥❛t✉r❛❧ ❢♦r❝✐♥❣ ❛❞❞✐♥❣ ❛ ♣s❡✉❞♦✐♥t❡rs❡❝t✐♦♥ ♦❢ F ✿ ✐❢ G ✐s ❛ M F ✲❣❡♥❡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❝✳ ✻ ✴ ✶✽

✳ ❊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 ♦❢ ✳ M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s ❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ω ω s✉❝❤ t❤❛t y ≤ ∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ω ω ✳ ✼ ✴ ✶✽

▼✐❧❧❡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 ♦❢ ✳ M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s ❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ω ω s✉❝❤ t❤❛t y ≤ ∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ω ω ✳ ✳ ❊①❛♠♣❧❡✿ ▲❛✈❡r ❢♦r❝✐♥❣✱ ❍❡❝❤❧❡r ❢♦r❝✐♥❣✳ ✼ ✴ ✶✽

❚❤❡♦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 ♦❢ ✳ M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s ❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ω ω s✉❝❤ t❤❛t y ≤ ∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ω ω ✳ ✳ ❊①❛♠♣❧❡✿ ▲❛✈❡r ❢♦r❝✐♥❣✱ ❍❡❝❤❧❡r ❢♦r❝✐♥❣✳ ▼✐❧❧❡r ❛♥❞ ❈♦❤❡♥ ❢♦r❝✐♥❣ ❞♦ ♥♦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 ♦❢ ✳ M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s ❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ω ω s✉❝❤ t❤❛t y ≤ ∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ω ω ✳ ✳ ❊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 M F ❞♦❡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 ♦❢ ✳ M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s ❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ω ω s✉❝❤ t❤❛t y ≤ ∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ω ω ✳ ✳ ❊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 M F ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✷ ✼ ✴ ✶✽

▲❡t ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ✳ ❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ✲❈❛♥❥❛r ✐❢ ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ ✳ M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s ❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ω ω s✉❝❤ t❤❛t y ≤ ∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ω ω ✳ ✳ ❊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 M F ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✷ ❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮ ❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ M F ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✼ ✴ ✶✽

❆ ✜❧t❡r ♦♥ ✐s ❝❛❧❧❡❞ ✲❈❛♥❥❛r ✐❢ ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ ✳ M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s ❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ω ω s✉❝❤ t❤❛t y ≤ ∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ω ω ✳ ✳ ❊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 M F ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✷ ❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮ ❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ M F ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t B ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ω ω ✳ ✼ ✴ ✶✽

M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s ❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ω ω s✉❝❤ t❤❛t y ≤ ∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ω ω ✳ ✳ ❊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 M F ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✷ ❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮ ❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ M F ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t B ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ω ω ✳ ❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ B ✲❈❛♥❥❛r ✐❢ M F ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ B ✳ ✷ ✼ ✴ ✶✽

M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s ❆ ♣♦s❡t P ✐s s❛✐❞ t♦ ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐❢ ✐♥ V P t❤❡r❡ ❡①✐sts x ∈ ω ω s✉❝❤ t❤❛t y ≤ ∗ x ❢♦r ❛❧❧ ❣r♦✉♥❞ ♠♦❞❡❧ y ∈ ω ω ✳ ✳ ❊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 M F ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ✷ ❉❡✜♥✐t✐♦♥ ✭●✉③♠❛♥✲❍r✉s❛❦✲▼❛rt✐♥❡③✮ ❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ ❈❛♥❥❛r ✐❢ M F ❞♦❡s ♥♦t ❛❞❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✳ ▲❡t B ❜❡ ❛♥ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ ω ω ✳ ❆ ✜❧t❡r F ♦♥ ω ✐s ❝❛❧❧❡❞ B ✲❈❛♥❥❛r ✐❢ M F ❛❞❞s ♥♦ r❡❛❧s ❞♦♠✐♥❛t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ B ✳ ✷ ✼ ✴ ✶✽

✷✮ ✳ ▲❡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❤❡♥ ✳ M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥ ❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮ ✶✮ ❊✈❡r② σ ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✽ ✴ ✶✽

❚❤❡♥ ❢♦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❤❡♥ ✳ M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥ ❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮ ✶✮ ❊✈❡r② σ ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ ( b = c ) ✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ✽ ✴ ✶✽

✐♥ 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❤❡♥ ✳ M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥ ❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮ ✶✮ ❊✈❡r② σ ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ ( b = c ) ✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ B = { b α : α < b } ⊂ ω ω s✉❝❤ t❤❛t b α ≤ ∗ b β ❢♦r ❛❧❧ α < β ✱ ✽ ✴ ✶✽

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

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

✱ t❤❡♥ ✳ M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥ ❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮ ✶✮ ❊✈❡r② σ ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ ( b = c ) ✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ B = { b α : α < b } ⊂ ω ω s✉❝❤ t❤❛t b α ≤ ∗ b β ❢♦r ❛❧❧ α < β ✱ ✐♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ b ♠❛♥② ❈♦❤❡♥s✱ t❤❡r❡ ❡①✐sts ❛ B ✲❈❛♥❥❛r F ⊃ F A ✳ ✷ ❚❤❡♦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✉❡♥❝❡ �C n : n ∈ ω � ♦❢ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ F + t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ � k n : n ∈ ω � ♦❢ ✐♥t❡❣❡rs s✉❝❤ t❤❛t ✐❢ X n ∈ C n ❢♦r ❛❧❧ n ✽ ✴ ✶✽

M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥ ❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮ ✶✮ ❊✈❡r② σ ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ ( b = c ) ✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ B = { b α : α < b } ⊂ ω ω s✉❝❤ t❤❛t b α ≤ ∗ b β ❢♦r ❛❧❧ α < β ✱ ✐♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ b ♠❛♥② ❈♦❤❡♥s✱ t❤❡r❡ ❡①✐sts ❛ B ✲❈❛♥❥❛r F ⊃ F A ✳ ✷ ❚❤❡♦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✉❡♥❝❡ �C n : n ∈ ω � ♦❢ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ F + t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ � k n : n ∈ ω � ♦❢ ✐♥t❡❣❡rs s✉❝❤ t❤❛t ✐❢ X n ∈ C n ❢♦r ❛❧❧ n ✱ t❤❡♥ n ∈ ω ( X n ∩ [ k n , k n +1 )) ∈ F + ✳ � ✽ ✴ ✶✽

M F ❛♥❞ ❞♦♠✐♥❛t✐♥❣ r❡❛❧s✿ ❝♦♥t✐♥✉❛t✐♦♥ ❚❤❡♦r❡♠ ✭❇r❡♥❞❧❡ ✶✾✾✽✮ ✶✮ ❊✈❡r② σ ✲❝♦♠♣❛❝t ✜❧t❡r ✐s ❈❛♥❥❛r✳ ✷✮ ( b = c ) ✳ ▲❡t A ❜❡ ❛ ♠❛❞ ❢❛♠✐❧②✳ ❚❤❡♥ ❢♦r ❛♥② ✉♥❜♦✉♥❞❡❞ B = { b α : α < b } ⊂ ω ω s✉❝❤ t❤❛t b α ≤ ∗ b β ❢♦r ❛❧❧ α < β ✱ ✐♥ t❤❡ ❡①t❡♥s✐♦♥ ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ b ♠❛♥② ❈♦❤❡♥s✱ t❤❡r❡ ❡①✐sts ❛ B ✲❈❛♥❥❛r F ⊃ F A ✳ ✷ ❚❤❡♦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✉❡♥❝❡ �C n : n ∈ ω � ♦❢ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ F + t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ � k n : n ∈ ω � ♦❢ ✐♥t❡❣❡rs s✉❝❤ t❤❛t ✐❢ X n ∈ C n ❢♦r ❛❧❧ n ✱ t❤❡♥ n ∈ ω ( X n ∩ [ k n , k n +1 )) ∈ F + ✳ � ✽ ✴ ✶✽

❆ 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 → ω ω ✳ ✾ ✴ ✶✽

❆ ✉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 → ω ω ✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f [ X ] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ω ω ✳ ✾ ✴ ✶✽

❆ ✉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 → ω ω ✳ ❆ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ▲✐♥❞❡❧ö❢ s♣❛❝❡ X ✐s ▼❡♥❣❡r ✐✛ f [ X ] ✐s ♥♦♥✲❞♦♠✐♥❛t✐♥❣ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s f : X → ω ω ✳ ✾ ✴ ✶✽

❚❤❡♦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 ✿✲✮ ✶✵ ✴ ✶✽

✷✮ ✐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 ✲❈❛♥❥❛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 ❛ ✷✮ M F ✐s ❛❧♠♦st ω ω ✲❜♦✉♥❞✐♥❣ s✉❜s♣❛❝❡ ♦❢ P ( ω ) ✳ ✶✵ ✴ ✶✽

✐✛ ✐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 ❛ ✷✮ M F ✐s ❛❧♠♦st ω ω ✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s s✉❜s♣❛❝❡ ♦❢ P ( ω ) ✳ B ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ω ω ✶✵ ✴ ✶✽

❈♦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 ❛ ✷✮ M F ✐s ❛❧♠♦st ω ω ✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s s✉❜s♣❛❝❡ ♦❢ P ( ω ) ✳ B ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ω ω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷ ✶✵ ✴ ✶✽

❚❤❡♥ ❞♦❡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 ❛ ✷✮ M F ✐s ❛❧♠♦st ω ω ✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s s✉❜s♣❛❝❡ ♦❢ P ( ω ) ✳ B ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ω ω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷ ❈♦r♦❧❧❛r② ▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω ✳ ✶✵ ✴ ✶✽

❆♥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 ❛ ✷✮ M F ✐s ❛❧♠♦st ω ω ✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s s✉❜s♣❛❝❡ ♦❢ P ( ω ) ✳ B ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ω ω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷ ❈♦r♦❧❧❛r② ▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω ✳ ❚❤❡♥ M F ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ F ✐s σ ✲❝♦♠♣❛❝t✳ ✷ ✶✵ ✴ ✶✽

❋♦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 ❛ ✷✮ M F ✐s ❛❧♠♦st ω ω ✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s s✉❜s♣❛❝❡ ♦❢ P ( ω ) ✳ B ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ω ω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷ ❈♦r♦❧❧❛r② ▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω ✳ ❚❤❡♥ M F ❞♦❡s ♥♦t ❛❞❞ ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✐✛ F ✐s σ ✲❝♦♠♣❛❝t✳ ✷ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❍r✉s❛❦ ❛♥❞ ▼✐♥❛♠✐✳ ✶✵ ✴ ✶✽

❈♦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 ❛ ✷✮ M F ✐s ❛❧♠♦st ω ω ✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s s✉❜s♣❛❝❡ ♦❢ P ( ω ) ✳ B ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ω ω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷ ❈♦r♦❧❧❛r② ▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω ✳ ❚❤❡♥ M F ❞♦❡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✳ ✶✵ ✴ ✶✽

❆♥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 ❛ ✷✮ M F ✐s ❛❧♠♦st ω ω ✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s s✉❜s♣❛❝❡ ♦❢ P ( ω ) ✳ B ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ω ω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷ ❈♦r♦❧❧❛r② ▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω ✳ ❚❤❡♥ M F ❞♦❡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 M F ( A ) ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ F ( A ) ✐s ♥♦t ❈❛♥❥❛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 ❛ ✷✮ M F ✐s ❛❧♠♦st ω ω ✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s s✉❜s♣❛❝❡ ♦❢ P ( ω ) ✳ B ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ω ω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷ ❈♦r♦❧❧❛r② ▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω ✳ ❚❤❡♥ M F ❞♦❡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 M F ( A ) ❛❞❞s ❛ ❞♦♠✐♥❛t✐♥❣ r❡❛❧ ✭❂ F ( A ) ✐s ♥♦t ❈❛♥❥❛r✮✳ ✷ ❆♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❇r❡♥❞❧❡✳ ✶✵ ✴ ✶✽

❚❤❡② ❛❧❧ st✉❞✐❡❞ ▼❡♥❣❡r ❛♥❞ ❍✉r❡✇✐❝③ ✜❧t❡rs ✿✲✮ ❚❤❡♦r❡♠ ✭❈❤♦❞♦✉♥s❦ý✲❘❡♣♦✈➨✲❩✳ ✷✵✶✹✮ ✶✮ F ✐s ❈❛♥❥❛r ✐✛ F ❤❛s t❤❡ ▼❡♥❣❡r ❝♦✈❡r✐♥❣ ♣r♦♣❡rt② ❛s ❛ ✷✮ M F ✐s ❛❧♠♦st ω ω ✲❜♦✉♥❞✐♥❣ ✐✛ F ✐s s✉❜s♣❛❝❡ ♦❢ P ( ω ) ✳ B ✲❈❛♥❥❛r ❢♦r ❛❧❧ ✉♥❜♦✉♥❞❡❞ B ⊂ ω ω ✐✛ F ✐s ❍✉r❡✇✐❝③✳ ✷ ❈♦r♦❧❧❛r② ▲❡t F ❜❡ ❛♥ ❛♥❛❧②t✐❝ ✜❧t❡r ♦♥ ω ✳ ❚❤❡♥ M F ❞♦❡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 M F ( 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✳ ✷ ✶✵ ✴ ✶✽

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 ) ✳ ✶✶ ✴ ✶✽

❚❤❡♦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❡♠ ✭❊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❡♠❛✐♥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✳ ✶✶ ✴ ✶✽

✷✳ ❜❡❝♦♠❡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❢✳ ✶✶ ✴ ✶✽

▼❡♥❣❡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✳ ✷ ✶✶ ✴ ✶✽

❚❤❡♥ ✐❢ 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✳ ✶✷ ✴ ✶✽

❚❤❡♦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❡♠ ✭●❛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❤❡♥ ω ω ≥ T K ( M ) ✐❢ ❛♥❞ ♦♥❧② ✐❢ M ✐s P♦❧✐s❤✳ ✶✷ ✴ ✶✽

◗✉❡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❤❡♥ ω ω ≥ 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❡✳ ✶✷ ✴ ✶✽

❨❡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❤❡♥ ω ω ≥ 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❤❄ ✶✷ ✴ ✶✽

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

❨❡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 ⊂ 2 ω 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 ⊂ 2 ω s✉❝❤ t❤❛t K ( Q ) ≥ T K ( X ) ❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮ ✭❱❂▲✮✳ ❚❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ X ⊂ 2 ω s✉❝❤ t❤❛t K ( Q ) > T K ( X ) ✳ ✶✸ ✴ ✶✽

❲❡ ✉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 ⊂ 2 ω s✉❝❤ t❤❛t K ( Q ) ≥ T K ( X ) ❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮ ✭❱❂▲✮✳ ❚❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ X ⊂ 2 ω s✉❝❤ t❤❛t K ( Q ) > T K ( X ) ✳ ❲❡ ❝♦♥str✉❝t ❛ ❝♦✲❛♥❛❧②t✐❝ ❍✉r❡✇✐❝③ Y ⊂ 2 ω s✉❝❤ t❤❛t X = 2 ω \ Y ✐s ❛s r❡q✉✐r❡❞✳ ✶✸ ✴ ✶✽

❍✉r❡✇✐❝③ s♣❛❝❡s ❛♥❞ ❚✉❦❡② ♦r❞❡r ◗✉❡st✐♦♥ ✭❋r❡♠❧✐♥ ✶✾✾✶✮ ■s ✐t ❝♦♥s✐st❡♥t t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ X ⊂ 2 ω s✉❝❤ t❤❛t K ( Q ) ≥ T K ( X ) ❄ ❨❡s✱ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❚❤❡♦r❡♠ ✭●❛rts✐❞❡✲▼❡❞✐♥✐✲❩✳ ✷✵✶✻✮ ✭❱❂▲✮✳ ❚❤❡r❡ ❡①✐sts ❛♥ ❛♥❛❧②t✐❝ ♥♦♥✲❇♦r❡❧ X ⊂ 2 ω s✉❝❤ t❤❛t K ( Q ) > T K ( X ) ✳ ❲❡ ❝♦♥str✉❝t ❛ ❝♦✲❛♥❛❧②t✐❝ ❍✉r❡✇✐❝③ Y ⊂ 2 ω s✉❝❤ t❤❛t X = 2 ω \ 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✳ ✶✸ ✴ ✶✽

■♥ 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 S 0 , S 1 ✇❤♦s❡ ♣r♦❞✉❝t ✐s ♥♦t ▼❡♥❣❡r✳ Pr♦♦❢✳ ❋✐① ❛ ❝♦✉♥t❛❜❧❡ ❞❡♥s❡ Q ⊂ 2 ω ❛♥❞ ✇r✐t❡ 2 ω \ Q = { x α : α < ω 1 } ✳ ✶✹ ✴ ✶✽

❤❛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 S 0 , S 1 ✇❤♦s❡ ♣r♦❞✉❝t ✐s ♥♦t ▼❡♥❣❡r✳ Pr♦♦❢✳ ❋✐① ❛ ❝♦✉♥t❛❜❧❡ ❞❡♥s❡ Q ⊂ 2 ω ❛♥❞ ✇r✐t❡ 2 ω \ Q = { x α : α < ω 1 } ✳ ■♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ❙✐❡r♣✐♥s❦✐ s❡t ❜② tr❛♥s✜♥✐t❡ ✐♥❞✉❝t✐♦♥ ❛t ❡❛❝❤ st❛❣❡ α ✇❡ ❝❛♥ ♣✐❝❦ ❛ ♣♦✐♥t s α ♦✉ts✐❞❡ ♦❢ ❛ ❣✐✈❡♥ ♠❡❛s✉r❡ ③❡r♦ s❡t Z α ⊂ 2 ω ✳ ✶✹ ✴ ✶✽

❈❤♦♦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 S 0 , S 1 ✇❤♦s❡ ♣r♦❞✉❝t ✐s ♥♦t ▼❡♥❣❡r✳ Pr♦♦❢✳ ❋✐① ❛ ❝♦✉♥t❛❜❧❡ ❞❡♥s❡ Q ⊂ 2 ω ❛♥❞ ✇r✐t❡ 2 ω \ Q = { x α : α < ω 1 } ✳ ■♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ❙✐❡r♣✐♥s❦✐ s❡t ❜② tr❛♥s✜♥✐t❡ ✐♥❞✉❝t✐♦♥ ❛t ❡❛❝❤ st❛❣❡ α ✇❡ ❝❛♥ ♣✐❝❦ ❛ ♣♦✐♥t s α ♦✉ts✐❞❡ ♦❢ ❛ ❣✐✈❡♥ ♠❡❛s✉r❡ ③❡r♦ s❡t Z α ⊂ 2 ω ✳ 2 ω ❤❛s ❛ ♥❛t✉r❛❧ str✉❝t✉r❡ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣✱ ❛♥❞ t❤❡ s✉♠ ♦❢ ❛♥② t✇♦ ♠❡❛s✉r❡ ✶ s❡ts ✐s t❤❡ ✇❤♦❧❡ ❣r♦✉♣✳ ✶✹ ✴ ✶✽

❙❡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 S 0 , S 1 ✇❤♦s❡ ♣r♦❞✉❝t ✐s ♥♦t ▼❡♥❣❡r✳ Pr♦♦❢✳ ❋✐① ❛ ❝♦✉♥t❛❜❧❡ ❞❡♥s❡ Q ⊂ 2 ω ❛♥❞ ✇r✐t❡ 2 ω \ Q = { x α : α < ω 1 } ✳ ■♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ❙✐❡r♣✐♥s❦✐ s❡t ❜② tr❛♥s✜♥✐t❡ ✐♥❞✉❝t✐♦♥ ❛t ❡❛❝❤ st❛❣❡ α ✇❡ ❝❛♥ ♣✐❝❦ ❛ ♣♦✐♥t s α ♦✉ts✐❞❡ ♦❢ ❛ ❣✐✈❡♥ ♠❡❛s✉r❡ ③❡r♦ s❡t Z α ⊂ 2 ω ✳ 2 ω ❤❛s ❛ ♥❛t✉r❛❧ str✉❝t✉r❡ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣✱ ❛♥❞ t❤❡ s✉♠ ♦❢ ❛♥② t✇♦ ♠❡❛s✉r❡ ✶ s❡ts ✐s t❤❡ α ∈ 2 ω \ Z α s✉❝❤ t❤❛t s 0 ✇❤♦❧❡ ❣r♦✉♣✳ ❈❤♦♦s❡ s 0 α , s 1 α + s 1 α = x α α + { s 1 − i ❛♥❞ s i : β < α } ∩ Q = ∅ ✳ β ✶✹ ✴ ✶✽