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■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ βω ✕ ❛ s♣❛❝❡ ♦❢ ✉❧tr❛✜❧t❡rs ♦♥ ω ✕ ❛♥❞ t❤❡ ❖♣❡♥ ❈♦❧♦✉r✐♥❣ ❆①✐♦♠ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s ▼❛② ✷✵✶✶ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ ❖✉t❧✐♥❡ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω ✸ βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

❈♦♥str✉❝t✐♦♥ ♦❢ ❲❡ ❞❡✜♥❡ t♦ ❜❡ t❤❡ ❙t♦♥❡ s♣❛❝❡ ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ ✱ t❤❛t ✐s✱ t❤❡ s♣❛❝❡ ♦❢ ✉❧tr❛✜❧t❡rs ♦♥ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❜❛s❡ ✱ ✇❤❡r❡ ❢♦r ❡❛❝❤ ✳ ✜♥ ❚❤❡ r❡♠❛✐♥❞❡r ✐s t❤❡ ❙t♦♥❡ s♣❛❝❡ ♦❢ t❤❡ q✉♦t✐❡♥t ✜♥ ✳ ■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ ■♥tr♦❞✉❝✐♥❣ t❤❡ ❝r❡❛t✉r❡✿ βω ❉❡✜♥✐t✐♦♥ ●✐✈❡♥ ❛ ✭❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❍❛✉s❞♦r✛✮ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✱ t❤❡ ❷❡❝❤✲❙t♦♥❡ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ X ✱ ✇r✐tt❡♥ βX ✱ ✐s ❛ ❝♦♠♣❛❝t s♣❛❝❡ s✉❝❤ t❤❛t ✶ X ✐s ❞❡♥s❡ ✐♥ βX ✷ ❋♦r ❡✈❡r② ❝♦♠♣❛❝t s♣❛❝❡ K ❛♥❞ ❡✈❡r② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ → K ✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ❡①t❡♥s✐♦♥ ♦❢ f ✱ f : X − → K ✳ βf : βX − ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

✜♥ ❚❤❡ r❡♠❛✐♥❞❡r ✐s t❤❡ ❙t♦♥❡ s♣❛❝❡ ♦❢ t❤❡ q✉♦t✐❡♥t ✜♥ ✳ ■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ ■♥tr♦❞✉❝✐♥❣ t❤❡ ❝r❡❛t✉r❡✿ βω ❉❡✜♥✐t✐♦♥ ●✐✈❡♥ ❛ ✭❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❍❛✉s❞♦r✛✮ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✱ t❤❡ ❷❡❝❤✲❙t♦♥❡ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ X ✱ ✇r✐tt❡♥ βX ✱ ✐s ❛ ❝♦♠♣❛❝t s♣❛❝❡ s✉❝❤ t❤❛t ✶ X ✐s ❞❡♥s❡ ✐♥ βX ✷ ❋♦r ❡✈❡r② ❝♦♠♣❛❝t s♣❛❝❡ K ❛♥❞ ❡✈❡r② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ → K ✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ❡①t❡♥s✐♦♥ ♦❢ f ✱ f : X − → K ✳ βf : βX − ❈♦♥str✉❝t✐♦♥ ♦❢ βω ❲❡ ❞❡✜♥❡ βω t♦ ❜❡ t❤❡ ❙t♦♥❡ s♣❛❝❡ ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ P ( ω ) ✱ t❤❛t ✐s✱ t❤❡ s♣❛❝❡ ♦❢ ✉❧tr❛✜❧t❡rs ♦♥ ω ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❜❛s❡ B = { B a : a ⊆ ω } ✱ ✇❤❡r❡ B a = { p ∈ βω : a ∈ p } ❢♦r ❡❛❝❤ a ⊆ ω ✳ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ ■♥tr♦❞✉❝✐♥❣ t❤❡ ❝r❡❛t✉r❡✿ βω ❉❡✜♥✐t✐♦♥ ●✐✈❡♥ ❛ ✭❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❍❛✉s❞♦r✛✮ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✱ t❤❡ ❷❡❝❤✲❙t♦♥❡ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ ♦❢ X ✱ ✇r✐tt❡♥ βX ✱ ✐s ❛ ❝♦♠♣❛❝t s♣❛❝❡ s✉❝❤ t❤❛t ✶ X ✐s ❞❡♥s❡ ✐♥ βX ✷ ❋♦r ❡✈❡r② ❝♦♠♣❛❝t s♣❛❝❡ K ❛♥❞ ❡✈❡r② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ → K ✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ❡①t❡♥s✐♦♥ ♦❢ f ✱ f : X − → K ✳ βf : βX − ❈♦♥str✉❝t✐♦♥ ♦❢ βω ❲❡ ❞❡✜♥❡ βω t♦ ❜❡ t❤❡ ❙t♦♥❡ s♣❛❝❡ ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ P ( ω ) ✱ t❤❛t ✐s✱ t❤❡ s♣❛❝❡ ♦❢ ✉❧tr❛✜❧t❡rs ♦♥ ω ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❜❛s❡ B = { B a : a ⊆ ω } ✱ ✇❤❡r❡ B a = { p ∈ βω : a ∈ p } ❢♦r ❡❛❝❤ a ⊆ ω ✳ P ( ω ) / ✜♥ ❚❤❡ r❡♠❛✐♥❞❡r βω \ ω = ω ∗ ✐s t❤❡ ❙t♦♥❡ s♣❛❝❡ ♦❢ t❤❡ q✉♦t✐❡♥t P ( ω ) / ✜♥ ✳ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

●❛♣s ✐♥ ✜♥ ■t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t ✲❣❛♣s ❞♦ ♥♦t ❡①✐st ✐♥ ✜♥ ✳ ❍♦✇❡✈❡r✱ ✲❣❛♣s ❞♦ ❡①✐st✳ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ βω ✉♥❞❡r ❖❈❆ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω ●❛♣s ❉❡✜♥✐t✐♦♥ ▲❡t ( B , < ) ❜❡ ❛ ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛✳ ❙✉♣♣♦s❡ A, B ⊆ B ❛r❡ ❧✐♥❡❛r❧② ♦r❞❡r❡❞ ❜② < ❛♥❞ ❤❛✈❡ ♦r❞❡r t②♣❡s λ ❛♥❞ κ r❡s♣❡❝t✐✈❡❧②✳ ❲❡ s❛② t❤❛t t❤❡ ♣❛✐r ( A, B ) ❢♦r♠s ❛ ( λ, κ ) ✲❣❛♣ ✐❢ a < b ❢♦r ❛❧❧ a ∈ A ❛♥❞ b ∈ B ❛♥❞ t❤❡r❡ ✐s ♥♦ c ∈ B s✉❝❤ t❤❛t a < c < b ❢♦r ❛❧❧ a ∈ A ❛♥❞ b ∈ B ✳ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ βω ✉♥❞❡r ❖❈❆ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω ●❛♣s ❉❡✜♥✐t✐♦♥ ▲❡t ( B , < ) ❜❡ ❛ ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛✳ ❙✉♣♣♦s❡ A, B ⊆ B ❛r❡ ❧✐♥❡❛r❧② ♦r❞❡r❡❞ ❜② < ❛♥❞ ❤❛✈❡ ♦r❞❡r t②♣❡s λ ❛♥❞ κ r❡s♣❡❝t✐✈❡❧②✳ ❲❡ s❛② t❤❛t t❤❡ ♣❛✐r ( A, B ) ❢♦r♠s ❛ ( λ, κ ) ✲❣❛♣ ✐❢ a < b ❢♦r ❛❧❧ a ∈ A ❛♥❞ b ∈ B ❛♥❞ t❤❡r❡ ✐s ♥♦ c ∈ B s✉❝❤ t❤❛t a < c < b ❢♦r ❛❧❧ a ∈ A ❛♥❞ b ∈ B ✳ ●❛♣s ✐♥ P ( ω ) / ✜♥ ■t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t ( ω, ω ) ✲❣❛♣s ❞♦ ♥♦t ❡①✐st ✐♥ P ( ω ) / ✜♥ ✳ ❍♦✇❡✈❡r✱ ( ω 1 , ω 1 ) ✲❣❛♣s ❞♦ ❡①✐st✳ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

❲✐t❤ t❤❡ ❝♦♥t✐♥✉✉♠ ❤②♣♦t❤❡s✐s ✇❡ ❣❡t✿ ❈♦r♦❧❛r② ❆ ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ ❡♠❜❡❞s ✐♥t♦ ✜♥ ✐✛ ✐t ✐s ♦❢ s✐③❡ ♦r ❧❡ss✳ ❚❤❡♦r❡♠ ✭P❛r♦✈✐↔❡♥❦♦✮ ❆ ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✜♥ ✐✛ t❤❡r❡ ❛r❡ ♥♦ ✲❣❛♣s ✐♥ ❛♥❞ ✳ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ βω ✉♥❞❡r ❖❈❆ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω ❆ ❝❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ P ( ω ) / ✜♥ ✉♥❞❡r ❈❍ ❚❤❡♦r❡♠ ✭P❛r♦✈✐↔❡♥❦♦✮ ❊✈❡r② ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ ♦❢ s✐③❡ ℵ 1 ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥t♦ P ( ω ) / ✜♥✳ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆