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Feb 23, 2024 123 likes •544 views

trt r t t tss r s trtrs t

slide-1
SLIDE 1

■♥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 ❛♥❞ ❖❈❆

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SLIDE 2

■♥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 ❛♥❞ ❖❈❆

slide-3
SLIDE 3

■♥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✐♦♥

f : X − → K✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ❡①t❡♥s✐♦♥ ♦❢ f✱ βf : βX − → K✳ ❈♦♥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 ✜♥✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-4
SLIDE 4

■♥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✐♦♥

f : X − → K✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ❡①t❡♥s✐♦♥ ♦❢ f✱ βf : βX − → K✳ ❈♦♥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 = {Ba : a ⊆ ω}✱ ✇❤❡r❡ Ba = {p ∈ βω : a ∈ p} ❢♦r ❡❛❝❤ a ⊆ ω✳ ✜♥ ❚❤❡ r❡♠❛✐♥❞❡r ✐s t❤❡ ❙t♦♥❡ s♣❛❝❡ ♦❢ t❤❡ q✉♦t✐❡♥t ✜♥✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

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SLIDE 5

■♥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✐♦♥

f : X − → K✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ❡①t❡♥s✐♦♥ ♦❢ f✱ βf : βX − → K✳ ❈♦♥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 = {Ba : a ⊆ ω}✱ ✇❤❡r❡ Ba = {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 ❛♥❞ ❖❈❆

slide-6
SLIDE 6

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜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 ✐♥

✜♥ ■t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t ✲❣❛♣s ❞♦ ♥♦t ❡①✐st ✐♥ ✜♥✳ ❍♦✇❡✈❡r✱ ✲❣❛♣s ❞♦ ❡①✐st✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-7
SLIDE 7

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜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✱ (ω1, ω1)✲❣❛♣s ❞♦ ❡①✐st✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-8
SLIDE 8

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❆ ❝❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ P(ω)/✜♥ ✉♥❞❡r ❈❍

❚❤❡♦r❡♠ ✭P❛r♦✈✐↔❡♥❦♦✮ ❊✈❡r② ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ ♦❢ s✐③❡ ℵ1 ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥t♦ P(ω)/✜♥✳ ❲✐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 ✐♥ ❛♥❞ ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-9
SLIDE 9

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❆ ❝❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ P(ω)/✜♥ ✉♥❞❡r ❈❍

❚❤❡♦r❡♠ ✭P❛r♦✈✐↔❡♥❦♦✮ ❊✈❡r② ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ ♦❢ s✐③❡ ℵ1 ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥t♦ P(ω)/✜♥✳ ❲✐t❤ t❤❡ ❝♦♥t✐♥✉✉♠ ❤②♣♦t❤❡s✐s ✇❡ ❣❡t✿ ❈♦r♦❧❛r② ❆ ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ ❡♠❜❡❞s ✐♥t♦ P(ω)/✜♥ ✐✛ ✐t ✐s ♦❢ s✐③❡ c ♦r ❧❡ss✳ ❚❤❡♦r❡♠ ✭P❛r♦✈✐↔❡♥❦♦✮ ❆ ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✜♥ ✐✛ t❤❡r❡ ❛r❡ ♥♦ ✲❣❛♣s ✐♥ ❛♥❞ ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-10
SLIDE 10

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❆ ❝❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ P(ω)/✜♥ ✉♥❞❡r ❈❍

❚❤❡♦r❡♠ ✭P❛r♦✈✐↔❡♥❦♦✮ ❊✈❡r② ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ ♦❢ s✐③❡ ℵ1 ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥t♦ P(ω)/✜♥✳ ❲✐t❤ t❤❡ ❝♦♥t✐♥✉✉♠ ❤②♣♦t❤❡s✐s ✇❡ ❣❡t✿ ❈♦r♦❧❛r② ❆ ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ ❡♠❜❡❞s ✐♥t♦ P(ω)/✜♥ ✐✛ ✐t ✐s ♦❢ s✐③❡ c ♦r ❧❡ss✳ ❚❤❡♦r❡♠ ✭P❛r♦✈✐↔❡♥❦♦✮ ❆ ❇♦♦❧❡❛♥ ❆❧❣❡❜r❛ B ✐s ✐s♦♠♦r♣❤✐❝ t♦ P(ω)/✜♥ ✐✛ t❤❡r❡ ❛r❡ ♥♦ (ω, ω)✲❣❛♣s ✐♥ B ❛♥❞ |B| ≤ c✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-11
SLIDE 11

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛

❉❡✜♥✐t✐♦♥ ❚❤❡ q✉♦t✐❡♥t ♦❢ t❤❡ σ✲❛❧❣❡❜r❛ ♦❢ ❇♦r❡❧ s❡ts ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ ❜② t❤❡ ✐❞❡❛❧ ♦❢ s❡ts ♦❢ ♠❡❛s✉r❡ ③❡r♦ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛✳ ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❛s ❛ s✉❜❛❧❣❡❜r❛ ♦❢ ✜♥ ❚❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ✐s ✳ ❍❡♥❝❡✱ ✉♥❞❡r ❈❍✱ t❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❡♠❜❡❞s ✐♥t♦ ✜♥✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-12
SLIDE 12

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛

❉❡✜♥✐t✐♦♥ ❚❤❡ q✉♦t✐❡♥t ♦❢ t❤❡ σ✲❛❧❣❡❜r❛ ♦❢ ❇♦r❡❧ s❡ts ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ ❜② t❤❡ ✐❞❡❛❧ ♦❢ s❡ts ♦❢ ♠❡❛s✉r❡ ③❡r♦ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛✳ ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❛s ❛ s✉❜❛❧❣❡❜r❛ ♦❢ P(ω)/✜♥ ❚❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ✐s c✳ ❍❡♥❝❡✱ ✉♥❞❡r ❈❍✱ t❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❡♠❜❡❞s ✐♥t♦ P(ω)/✜♥✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-13
SLIDE 13

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❆✉t♦♠♦r♣❤✐s♠s ♦❢ P(ω)/✜♥

❚r✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❆♥ ❛✉t♦♠♦r♣❤✐s♠ ϕ ♦❢ P(ω)/✜♥ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ tr✐✈✐❛❧ ✐❢ ✐t ✐s ✐♥❞✉❝❡❞ ❜② ❛♥ ❛❧♠♦st ♣❡r♠✉t❛t✐♦♥ ♦❢ ω✱ ✐✳❡✳✱ ✐❢ t❤❡r❡ ❡①✐st ✜♥✐t❡ s❡ts a✱ b ⊆ ω ❛♥❞ ❛ ❜✐❥❡❝t✐♦♥ e : ω \ a − → ω \ b s✉❝❤ t❤❛t ❢♦r ❡✈❡r② x ⊆ ω ✇❡ ❤❛✈❡ ϕ([x]) = [e′′(x)]✳ ❚❤❡r❡ ❛r❡ ♥♦♥✲tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❯♥❞❡r ❈❍✱ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ✱ ✇❤✐❝❤ ❤❛s ❛✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s✳ ❙♦ ✜♥ ❤❛s ❛t ❧❡❛st ❛✉t♦♠♦r♣❤✐s♠s✳ ❈❧❡❛r❧②✱ ✐t ❝❛♥♥♦t ❤❛✈❡ ♠♦r❡✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-14
SLIDE 14

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❆✉t♦♠♦r♣❤✐s♠s ♦❢ P(ω)/✜♥

❚r✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❆♥ ❛✉t♦♠♦r♣❤✐s♠ ϕ ♦❢ P(ω)/✜♥ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ tr✐✈✐❛❧ ✐❢ ✐t ✐s ✐♥❞✉❝❡❞ ❜② ❛♥ ❛❧♠♦st ♣❡r♠✉t❛t✐♦♥ ♦❢ ω✱ ✐✳❡✳✱ ✐❢ t❤❡r❡ ❡①✐st ✜♥✐t❡ s❡ts a✱ b ⊆ ω ❛♥❞ ❛ ❜✐❥❡❝t✐♦♥ e : ω \ a − → ω \ b s✉❝❤ t❤❛t ❢♦r ❡✈❡r② x ⊆ ω ✇❡ ❤❛✈❡ ϕ([x]) = [e′′(x)]✳ ❚❤❡r❡ ❛r❡ ♥♦♥✲tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❯♥❞❡r ❈❍✱ ω∗ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ω × 2c✱ ✇❤✐❝❤ ❤❛s 2c ❛✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s✳ ❙♦ P(ω)/✜♥ ❤❛s ❛t ❧❡❛st 2c ❛✉t♦♠♦r♣❤✐s♠s✳ ❈❧❡❛r❧②✱ ✐t ❝❛♥♥♦t ❤❛✈❡ ♠♦r❡✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-15
SLIDE 15

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❚❤❡ ❖♣❡♥ ❈♦❧♦✉r✐♥❣ ❆①✐♦♠

❉❡✜♥✐t✐♦♥ ▲❡t X ❜❡ ❛ s✉❜s❡t ♦❢ P(ω) ✭♦r ♦❢ ❛♥② s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✮✳ ❲❡ ✇✐❧❧ ❝❛❧❧ K ⊆ [X]2 = {x ⊆ X : |x| = 2} ♦♣❡♥ ✐♥ [X]2 ✐❢ t❤❡ s❡t {a, b : {a, b} ∈ K} ✐s ♦♣❡♥ ✐♥ t❤❡ s♣❛❝❡ X × X ✇✐t❤ t❤❡ ♣r♦❞✉❝t t♦♣♦❧♦❣②✳ ❖❈❆ ■❢ ✐s ❛ s✉❜s❡t ♦❢ ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡ s♣❛❝❡ ❛♥❞ ✐❢ ✐s ❛ ♣❛rt✐t✐♦♥ s✉❝❤ t❤❛t ✐s ♦♣❡♥ ✐♥ ✱ t❤❡♥

✶ ❡✐t❤❡r t❤❡r❡ ❡①✐sts ❛♥ ✉♥❝♦✉♥t❛❜❧❡

✇✐t❤

✷ ♦r t❤❡r❡ ❡①✐st s❡ts

✱ ✱ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ❛♥❞ ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-16
SLIDE 16

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❚❤❡ ❖♣❡♥ ❈♦❧♦✉r✐♥❣ ❆①✐♦♠

❉❡✜♥✐t✐♦♥ ▲❡t X ❜❡ ❛ s✉❜s❡t ♦❢ P(ω) ✭♦r ♦❢ ❛♥② s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✮✳ ❲❡ ✇✐❧❧ ❝❛❧❧ K ⊆ [X]2 = {x ⊆ X : |x| = 2} ♦♣❡♥ ✐♥ [X]2 ✐❢ t❤❡ s❡t {a, b : {a, b} ∈ K} ✐s ♦♣❡♥ ✐♥ t❤❡ s♣❛❝❡ X × X ✇✐t❤ t❤❡ ♣r♦❞✉❝t t♦♣♦❧♦❣②✳ ❖❈❆ ■❢ X ✐s ❛ s✉❜s❡t ♦❢ ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡ s♣❛❝❡ ❛♥❞ ✐❢ [X]2 = K0 ∪ K1 ✐s ❛ ♣❛rt✐t✐♦♥ s✉❝❤ t❤❛t K0 ✐s ♦♣❡♥ ✐♥ [X]2✱ t❤❡♥

✶ ❡✐t❤❡r t❤❡r❡ ❡①✐sts ❛♥ ✉♥❝♦✉♥t❛❜❧❡ Y ⊆ X ✇✐t❤ [Y ]2 ⊆ K0 ✷ ♦r t❤❡r❡ ❡①✐st s❡ts Hn ⊆ X✱ n < ω✱ s✉❝❤ t❤❛t [Hn]2 ⊆ K1 ❢♦r

❛❧❧ n < ω ❛♥❞ X =

n<ω Hn✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-17
SLIDE 17

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛

❚❤❡♦r❡♠ ✭❉♦✇✲❍❛rt✮ ✭❖❈❆✮ ❚❤❡ ♠❡s✉r❡ ❛❧❣❡❜r❛ ❝❛♥♥♦t ❜❡ ❡♠❜❡❞❞❡❞ ✐♥t♦ P(ω)/✜♥✳ Pr♦♦❢✳

✶ ❲❡ st❛rt ❜② s❤♦✇✐♥❣ t❤❛t ❖❈❆ ✐♠♣❧✐❡s t❤❛t ❛♥ ❡♠❜❡❞❞✐♥❣

♦❢ t❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ✐♥t♦ ✜♥ ♠✉st ❤❛✈❡ ❛ s♣❡❝✐❛❧ str✉❝t✉r❡✳

✷ ❍♦✇❡✈❡r✱ ✐t ❝❛♥ ❜❡ s❤♦✇♥ ✐♥ ❩❋❈ t❤❛t s✉❝❤ ❛♥ ❡♠❜❡❞❞✐♥❣

❝❛♥♥♦t ❤❛✈❡ t❤✐s str✉❝t✉r❡✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-18
SLIDE 18

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛

❚❤❡♦r❡♠ ✭❉♦✇✲❍❛rt✮ ✭❖❈❆✮ ❚❤❡ ♠❡s✉r❡ ❛❧❣❡❜r❛ ❝❛♥♥♦t ❜❡ ❡♠❜❡❞❞❡❞ ✐♥t♦ P(ω)/✜♥✳ Pr♦♦❢✳

✶ ❲❡ st❛rt ❜② s❤♦✇✐♥❣ t❤❛t ❖❈❆ ✐♠♣❧✐❡s t❤❛t ❛♥ ❡♠❜❡❞❞✐♥❣ ϕ ♦❢

t❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ✐♥t♦ P(ω)/✜♥ ♠✉st ❤❛✈❡ ❛ s♣❡❝✐❛❧ str✉❝t✉r❡✳

✷ ❍♦✇❡✈❡r✱ ✐t ❝❛♥ ❜❡ s❤♦✇♥ ✐♥ ❩❋❈ t❤❛t s✉❝❤ ❛♥ ❡♠❜❡❞❞✐♥❣

❝❛♥♥♦t ❤❛✈❡ t❤✐s str✉❝t✉r❡✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-19
SLIDE 19

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛

❚❤❡♦r❡♠ ✭❉♦✇✲❍❛rt✮ ✭❖❈❆✮ ❚❤❡ ♠❡s✉r❡ ❛❧❣❡❜r❛ ❝❛♥♥♦t ❜❡ ❡♠❜❡❞❞❡❞ ✐♥t♦ P(ω)/✜♥✳ Pr♦♦❢✳

✶ ❲❡ st❛rt ❜② s❤♦✇✐♥❣ t❤❛t ❖❈❆ ✐♠♣❧✐❡s t❤❛t ❛♥ ❡♠❜❡❞❞✐♥❣ ϕ ♦❢

t❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ✐♥t♦ P(ω)/✜♥ ♠✉st ❤❛✈❡ ❛ s♣❡❝✐❛❧ str✉❝t✉r❡✳

✷ ❍♦✇❡✈❡r✱ ✐t ❝❛♥ ❜❡ s❤♦✇♥ ✐♥ ❩❋❈ t❤❛t s✉❝❤ ❛♥ ❡♠❜❡❞❞✐♥❣

❝❛♥♥♦t ❤❛✈❡ t❤✐s str✉❝t✉r❡✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-20
SLIDE 20

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❚❤❡♦r❡♠ ✭❱❡❧✐↔❦♦✈✐➣✮ ✭❖❈❆ ✰ ▼❆ℵ1✮ ❊✈❡r② ❛✉t♦♠♦r♣❤✐s♠ ♦❢ P(ω)/✜♥ ✐s tr✐✈✐❛❧✳ ❲❡ ♥❡❡❞ ❛ ❝♦✉♣❧❡ ♦❢ ❞❡✜♥✐t✐♦♥s✿ ❉❡✜♥✐t✐♦♥ ❆♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ✜♥ ✐s tr✐✈✐❛❧ ♦♥ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t ✱ ❢♦r ❛❧❧ ❉❡✜♥✐t✐♦♥ ❆ ❢❛♠✐❧② ♦❢ ✐♥✜♥✐t❡ s❡ts ✐s ❝❛❧❧❡❞ ❛❧♠♦st ❞✐s❥♦✐♥t ✐❢ ❢♦r ❛❧❧ ✱ ✱ ✐s ✜♥✐t❡✳ ❆♥ ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ✐s ❝❛❧❧❡❞ ♥❡❛t ✐❢ t❤❡r❡ ✐s ❛ ✶✲✶ ♠❛♣ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ✱ ✐s ❛♥ ✐♥✜♥✐t❡ ❜r❛♥❝❤ t❤r♦✉❣❤ ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-21
SLIDE 21

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❚❤❡♦r❡♠ ✭❱❡❧✐↔❦♦✈✐➣✮ ✭❖❈❆ ✰ ▼❆ℵ1✮ ❊✈❡r② ❛✉t♦♠♦r♣❤✐s♠ ♦❢ P(ω)/✜♥ ✐s tr✐✈✐❛❧✳ ❲❡ ♥❡❡❞ ❛ ❝♦✉♣❧❡ ♦❢ ❞❡✜♥✐t✐♦♥s✿ ❉❡✜♥✐t✐♦♥ ❆♥ ❛✉t♦♠♦r♣❤✐s♠ ϕ ♦❢ P(ω)/✜♥ ✐s tr✐✈✐❛❧ ♦♥ a ⊆ ω ✐❢ t❤❡r❡ ❡①✐sts ❛ ❢✉♥❝t✐♦♥ e : a → ω s✉❝❤ t❤❛t ϕ([x]) = [e′′(x)]✱ ❢♦r ❛❧❧ x ⊆ a ❉❡✜♥✐t✐♦♥ ❆ ❢❛♠✐❧② ♦❢ ✐♥✜♥✐t❡ s❡ts ✐s ❝❛❧❧❡❞ ❛❧♠♦st ❞✐s❥♦✐♥t ✐❢ ❢♦r ❛❧❧ ✱ ✱ ✐s ✜♥✐t❡✳ ❆♥ ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ✐s ❝❛❧❧❡❞ ♥❡❛t ✐❢ t❤❡r❡ ✐s ❛ ✶✲✶ ♠❛♣ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ✱ ✐s ❛♥ ✐♥✜♥✐t❡ ❜r❛♥❝❤ t❤r♦✉❣❤ ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-22
SLIDE 22

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❚❤❡♦r❡♠ ✭❱❡❧✐↔❦♦✈✐➣✮ ✭❖❈❆ ✰ ▼❆ℵ1✮ ❊✈❡r② ❛✉t♦♠♦r♣❤✐s♠ ♦❢ P(ω)/✜♥ ✐s tr✐✈✐❛❧✳ ❲❡ ♥❡❡❞ ❛ ❝♦✉♣❧❡ ♦❢ ❞❡✜♥✐t✐♦♥s✿ ❉❡✜♥✐t✐♦♥ ❆♥ ❛✉t♦♠♦r♣❤✐s♠ ϕ ♦❢ P(ω)/✜♥ ✐s tr✐✈✐❛❧ ♦♥ a ⊆ ω ✐❢ t❤❡r❡ ❡①✐sts ❛ ❢✉♥❝t✐♦♥ e : a → ω s✉❝❤ t❤❛t ϕ([x]) = [e′′(x)]✱ ❢♦r ❛❧❧ x ⊆ a ❉❡✜♥✐t✐♦♥ ❆ ❢❛♠✐❧② A ♦❢ ✐♥✜♥✐t❡ s❡ts ✐s ❝❛❧❧❡❞ ❛❧♠♦st ❞✐s❥♦✐♥t ✐❢ ❢♦r ❛❧❧ a✱ b ∈ A✱ a ∩ b ✐s ✜♥✐t❡✳ ❆♥ ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② A ♦❢ s✉❜s❡ts ♦❢ ω ✐s ❝❛❧❧❡❞ ♥❡❛t ✐❢ t❤❡r❡ ✐s ❛ ✶✲✶ ♠❛♣ e : ω → 2<ω s✉❝❤ t❤❛t ❢♦r ❡✈❡r② a ∈ A✱ ∪e′′(a) ✐s ❛♥ ✐♥✜♥✐t❡ ❜r❛♥❝❤ t❤r♦✉❣❤ 2<ω✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-23
SLIDE 23

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❆ s❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢

❲❡ ✜① ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ϕ ♦❢ P(ω)/✜♥✳ ▲❡♠♠❛ ▲❡t A ❜❡ ❛ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ω✳ ❚❤❡♥✱ ϕ ✐s tr✐✈✐❛❧ ♦♥ ❛❧❧ ❜✉t ❝♦✉♥t❛❜❧② ♠❛♥② c ∈ A✳ ▲❡♠♠❛ ■❢ ✐s ♥♦♥tr✐✈✐❛❧✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ✉♥❝♦✉♥t❛❜❧❡ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② s✉❝❤ t❤❛t ✐s ♥♦♥tr✐✈✐❛❧ ♦♥ ❡✈❡r② ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-24
SLIDE 24

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❆ s❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢

❲❡ ✜① ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ϕ ♦❢ P(ω)/✜♥✳ ▲❡♠♠❛ ▲❡t A ❜❡ ❛ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ω✳ ❚❤❡♥✱ ϕ ✐s tr✐✈✐❛❧ ♦♥ ❛❧❧ ❜✉t ❝♦✉♥t❛❜❧② ♠❛♥② c ∈ A✳ ▲❡♠♠❛ ■❢ ϕ ✐s ♥♦♥tr✐✈✐❛❧✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ✉♥❝♦✉♥t❛❜❧❡ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② A s✉❝❤ t❤❛t ϕ ✐s ♥♦♥tr✐✈✐❛❧ ♦♥ ❡✈❡r② a ∈ A✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-25
SLIDE 25

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

▲❡♠♠❛ ▲❡t A ❜❡ ❛ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ω✳ ❚❤❡♥✱ ϕ ✐s tr✐✈✐❛❧ ♦♥ ❛❧❧ ❜✉t ❝♦✉♥t❛❜❧② ♠❛♥② c ∈ A✳ Pr♦♦❢

✶ ❋✐①

s✉❝❤ t❤❛t ✱ ❢♦r ❡✈❡r② ✳

✷ ❋✐①

✇✐t♥❡ss✐♥❣ t❤❡ ❢❛❝t t❤❛t ✐s ♥❡❛t✳

✸ ❉❡✜♥❡

t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢ s✉❜s❡ts ♦❢ s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ✇❤✐❝❤ s❛t✐s✜❡s ✳

✹ ❉❡✜♥❡ t❤❡ ♣❛rt✐t✐♦♥

❜②✿ ✐✛

✶ ✷ ✸ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-26
SLIDE 26

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

▲❡♠♠❛ ▲❡t A ❜❡ ❛ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ω✳ ❚❤❡♥✱ ϕ ✐s tr✐✈✐❛❧ ♦♥ ❛❧❧ ❜✉t ❝♦✉♥t❛❜❧② ♠❛♥② c ∈ A✳ Pr♦♦❢

✶ ❋✐① F : P(ω) −

→ P(ω) s✉❝❤ t❤❛t ϕ([x]) = [F(x)]✱ ❢♦r ❡✈❡r② x ⊆ ω✳

✷ ❋✐①

✇✐t♥❡ss✐♥❣ t❤❡ ❢❛❝t t❤❛t ✐s ♥❡❛t✳

✸ ❉❡✜♥❡

t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢ s✉❜s❡ts ♦❢ s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ✇❤✐❝❤ s❛t✐s✜❡s ✳

✹ ❉❡✜♥❡ t❤❡ ♣❛rt✐t✐♦♥

❜②✿ ✐✛

✶ ✷ ✸ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-27
SLIDE 27

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

▲❡♠♠❛ ▲❡t A ❜❡ ❛ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ω✳ ❚❤❡♥✱ ϕ ✐s tr✐✈✐❛❧ ♦♥ ❛❧❧ ❜✉t ❝♦✉♥t❛❜❧② ♠❛♥② c ∈ A✳ Pr♦♦❢

✶ ❋✐① F : P(ω) −

→ P(ω) s✉❝❤ t❤❛t ϕ([x]) = [F(x)]✱ ❢♦r ❡✈❡r② x ⊆ ω✳

✷ ❋✐① e : ω → 2<ω ✇✐t♥❡ss✐♥❣ t❤❡ ❢❛❝t t❤❛t A ✐s ♥❡❛t✳ ✸ ❉❡✜♥❡

t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢ s✉❜s❡ts ♦❢ s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ✇❤✐❝❤ s❛t✐s✜❡s ✳

✹ ❉❡✜♥❡ t❤❡ ♣❛rt✐t✐♦♥

❜②✿ ✐✛

✶ ✷ ✸ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-28
SLIDE 28

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

▲❡♠♠❛ ▲❡t A ❜❡ ❛ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ω✳ ❚❤❡♥✱ ϕ ✐s tr✐✈✐❛❧ ♦♥ ❛❧❧ ❜✉t ❝♦✉♥t❛❜❧② ♠❛♥② c ∈ A✳ Pr♦♦❢

✶ ❋✐① F : P(ω) −

→ P(ω) s✉❝❤ t❤❛t ϕ([x]) = [F(x)]✱ ❢♦r ❡✈❡r② x ⊆ ω✳

✷ ❋✐① e : ω → 2<ω ✇✐t♥❡ss✐♥❣ t❤❡ ❢❛❝t t❤❛t A ✐s ♥❡❛t✳ ✸ ❉❡✜♥❡ X t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs a, b ♦❢ s✉❜s❡ts ♦❢ ω s✉❝❤

t❤❛t t❤❡r❡ ❡①✐sts c ∈ A ✇❤✐❝❤ s❛t✐s✜❡s b ⊆ a ⊆ c✳

✹ ❉❡✜♥❡ t❤❡ ♣❛rt✐t✐♦♥

❜②✿ ✐✛

✶ ✷ ✸ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-29
SLIDE 29

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

▲❡♠♠❛ ▲❡t A ❜❡ ❛ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ω✳ ❚❤❡♥✱ ϕ ✐s tr✐✈✐❛❧ ♦♥ ❛❧❧ ❜✉t ❝♦✉♥t❛❜❧② ♠❛♥② c ∈ A✳ Pr♦♦❢

✶ ❋✐① F : P(ω) −

→ P(ω) s✉❝❤ t❤❛t ϕ([x]) = [F(x)]✱ ❢♦r ❡✈❡r② x ⊆ ω✳

✷ ❋✐① e : ω → 2<ω ✇✐t♥❡ss✐♥❣ t❤❡ ❢❛❝t t❤❛t A ✐s ♥❡❛t✳ ✸ ❉❡✜♥❡ X t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs a, b ♦❢ s✉❜s❡ts ♦❢ ω s✉❝❤

t❤❛t t❤❡r❡ ❡①✐sts c ∈ A ✇❤✐❝❤ s❛t✐s✜❡s b ⊆ a ⊆ c✳

✹ ❉❡✜♥❡ t❤❡ ♣❛rt✐t✐♦♥ [X]2 = K0 ∪ K1 ❜②✿ {a, b, a, b} ∈ K0

✐✛

✶ ✷ ✸ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-30
SLIDE 30

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

▲❡♠♠❛ ▲❡t A ❜❡ ❛ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ω✳ ❚❤❡♥✱ ϕ ✐s tr✐✈✐❛❧ ♦♥ ❛❧❧ ❜✉t ❝♦✉♥t❛❜❧② ♠❛♥② c ∈ A✳ Pr♦♦❢

✶ ❋✐① F : P(ω) −

→ P(ω) s✉❝❤ t❤❛t ϕ([x]) = [F(x)]✱ ❢♦r ❡✈❡r② x ⊆ ω✳

✷ ❋✐① e : ω → 2<ω ✇✐t♥❡ss✐♥❣ t❤❡ ❢❛❝t t❤❛t A ✐s ♥❡❛t✳ ✸ ❉❡✜♥❡ X t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs a, b ♦❢ s✉❜s❡ts ♦❢ ω s✉❝❤

t❤❛t t❤❡r❡ ❡①✐sts c ∈ A ✇❤✐❝❤ s❛t✐s✜❡s b ⊆ a ⊆ c✳

✹ ❉❡✜♥❡ t❤❡ ♣❛rt✐t✐♦♥ [X]2 = K0 ∪ K1 ❜②✿ {a, b, a, b} ∈ K0

✐✛

✶

∪e′′(a) = ∪e′′(a)

✷ ✸ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-31
SLIDE 31

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

▲❡♠♠❛ ▲❡t A ❜❡ ❛ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ω✳ ❚❤❡♥✱ ϕ ✐s tr✐✈✐❛❧ ♦♥ ❛❧❧ ❜✉t ❝♦✉♥t❛❜❧② ♠❛♥② c ∈ A✳ Pr♦♦❢

✶ ❋✐① F : P(ω) −

→ P(ω) s✉❝❤ t❤❛t ϕ([x]) = [F(x)]✱ ❢♦r ❡✈❡r② x ⊆ ω✳

✷ ❋✐① e : ω → 2<ω ✇✐t♥❡ss✐♥❣ t❤❡ ❢❛❝t t❤❛t A ✐s ♥❡❛t✳ ✸ ❉❡✜♥❡ X t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs a, b ♦❢ s✉❜s❡ts ♦❢ ω s✉❝❤

t❤❛t t❤❡r❡ ❡①✐sts c ∈ A ✇❤✐❝❤ s❛t✐s✜❡s b ⊆ a ⊆ c✳

✹ ❉❡✜♥❡ t❤❡ ♣❛rt✐t✐♦♥ [X]2 = K0 ∪ K1 ❜②✿ {a, b, a, b} ∈ K0

✐✛

✶

∪e′′(a) = ∪e′′(a)

✷

a ∩ b = a ∩ b

✸ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-32
SLIDE 32

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

▲❡♠♠❛ ▲❡t A ❜❡ ❛ ♥❡❛t ❛❧♠♦st ❞✐s❥♦✐♥t ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ ω✳ ❚❤❡♥✱ ϕ ✐s tr✐✈✐❛❧ ♦♥ ❛❧❧ ❜✉t ❝♦✉♥t❛❜❧② ♠❛♥② c ∈ A✳ Pr♦♦❢

✶ ❋✐① F : P(ω) −

→ P(ω) s✉❝❤ t❤❛t ϕ([x]) = [F(x)]✱ ❢♦r ❡✈❡r② x ⊆ ω✳

✷ ❋✐① e : ω → 2<ω ✇✐t♥❡ss✐♥❣ t❤❡ ❢❛❝t t❤❛t A ✐s ♥❡❛t✳ ✸ ❉❡✜♥❡ X t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs a, b ♦❢ s✉❜s❡ts ♦❢ ω s✉❝❤

t❤❛t t❤❡r❡ ❡①✐sts c ∈ A ✇❤✐❝❤ s❛t✐s✜❡s b ⊆ a ⊆ c✳

✹ ❉❡✜♥❡ t❤❡ ♣❛rt✐t✐♦♥ [X]2 = K0 ∪ K1 ❜②✿ {a, b, a, b} ∈ K0

✐✛

✶

∪e′′(a) = ∪e′′(a)

✷

a ∩ b = a ∩ b

✸

F(a) ∩ F(b) = F(a) ∩ F(b)

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-33
SLIDE 33

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

Pr♦♦❢ ✭❝♦♥t✐♥✉❡❞✮

✶ ❈♦♥s✐❞❡r t❤❡ t♦♣♦❧♦❣② τ ♦♥ X ♦❜t❛✐♥❡❞ ❜② ✐❞❡♥t✐❢②✐♥❣ a, b

✇✐t❤ a, b, F(a), F(b) ∈ P(ω)4✳ ❚❤❡♥✱ (X, τ) ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡✳

✷ ❙❤♦✇ t❤❛t

✐s ♦♣❡♥ ✐♥ s♦ t❤❛t ✇❡ ❝❛♥ ❝❛❧❧ ✉♣♦♥ ❖❈❆✳

✸ ❚❤❡r❡ ✐s ♥♦ ✉♥❝♦✉♥t❛❜❧❡

s✉❝❤ t❤❛t ✳

✹ ❇② ❖❈❆ ♦❜t❛✐♥ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥

✱ s✉❝❤ t❤❛t ✱ ❢♦r ❛❧❧ ✳

✺ ❋♦r ❡✈❡r②

❝❤♦♦s❡ ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t ❞❡♥s❡ ✐♥ ✇✐t❤ r❡s♣❡❝t t♦ ✳

✻ ❈❤♦♦s❡ ❛

❢♦r ❡✈❡r② s✉❝❤ t❤❛t ✱ ❛♥❞ ❞❡✜♥❡

✼ ❙❤♦✇ t❤❛t

✐s tr✐✈✐❛❧ ♦♥ ❡✈❡r② ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-34
SLIDE 34

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

Pr♦♦❢ ✭❝♦♥t✐♥✉❡❞✮

✶ ❈♦♥s✐❞❡r t❤❡ t♦♣♦❧♦❣② τ ♦♥ X ♦❜t❛✐♥❡❞ ❜② ✐❞❡♥t✐❢②✐♥❣ a, b

✇✐t❤ a, b, F(a), F(b) ∈ P(ω)4✳ ❚❤❡♥✱ (X, τ) ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡✳

✷ ❙❤♦✇ t❤❛t K0 ✐s ♦♣❡♥ ✐♥ τ s♦ t❤❛t ✇❡ ❝❛♥ ❝❛❧❧ ✉♣♦♥ ❖❈❆✳ ✸ ❚❤❡r❡ ✐s ♥♦ ✉♥❝♦✉♥t❛❜❧❡

s✉❝❤ t❤❛t ✳

✹ ❇② ❖❈❆ ♦❜t❛✐♥ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥

✱ s✉❝❤ t❤❛t ✱ ❢♦r ❛❧❧ ✳

✺ ❋♦r ❡✈❡r②

❝❤♦♦s❡ ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t ❞❡♥s❡ ✐♥ ✇✐t❤ r❡s♣❡❝t t♦ ✳

✻ ❈❤♦♦s❡ ❛

❢♦r ❡✈❡r② s✉❝❤ t❤❛t ✱ ❛♥❞ ❞❡✜♥❡

✼ ❙❤♦✇ t❤❛t

✐s tr✐✈✐❛❧ ♦♥ ❡✈❡r② ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-35
SLIDE 35

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

Pr♦♦❢ ✭❝♦♥t✐♥✉❡❞✮

✶ ❈♦♥s✐❞❡r t❤❡ t♦♣♦❧♦❣② τ ♦♥ X ♦❜t❛✐♥❡❞ ❜② ✐❞❡♥t✐❢②✐♥❣ a, b

✇✐t❤ a, b, F(a), F(b) ∈ P(ω)4✳ ❚❤❡♥✱ (X, τ) ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡✳

✷ ❙❤♦✇ t❤❛t K0 ✐s ♦♣❡♥ ✐♥ τ s♦ t❤❛t ✇❡ ❝❛♥ ❝❛❧❧ ✉♣♦♥ ❖❈❆✳ ✸ ❚❤❡r❡ ✐s ♥♦ ✉♥❝♦✉♥t❛❜❧❡ Y ⊆ X s✉❝❤ t❤❛t [Y ]2 ⊆ K0✳ ✹ ❇② ❖❈❆ ♦❜t❛✐♥ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥

✱ s✉❝❤ t❤❛t ✱ ❢♦r ❛❧❧ ✳

✺ ❋♦r ❡✈❡r②

❝❤♦♦s❡ ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t ❞❡♥s❡ ✐♥ ✇✐t❤ r❡s♣❡❝t t♦ ✳

✻ ❈❤♦♦s❡ ❛

❢♦r ❡✈❡r② s✉❝❤ t❤❛t ✱ ❛♥❞ ❞❡✜♥❡

✼ ❙❤♦✇ t❤❛t

✐s tr✐✈✐❛❧ ♦♥ ❡✈❡r② ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-36
SLIDE 36

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

Pr♦♦❢ ✭❝♦♥t✐♥✉❡❞✮

✶ ❈♦♥s✐❞❡r t❤❡ t♦♣♦❧♦❣② τ ♦♥ X ♦❜t❛✐♥❡❞ ❜② ✐❞❡♥t✐❢②✐♥❣ a, b

✇✐t❤ a, b, F(a), F(b) ∈ P(ω)4✳ ❚❤❡♥✱ (X, τ) ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡✳

✷ ❙❤♦✇ t❤❛t K0 ✐s ♦♣❡♥ ✐♥ τ s♦ t❤❛t ✇❡ ❝❛♥ ❝❛❧❧ ✉♣♦♥ ❖❈❆✳ ✸ ❚❤❡r❡ ✐s ♥♦ ✉♥❝♦✉♥t❛❜❧❡ Y ⊆ X s✉❝❤ t❤❛t [Y ]2 ⊆ K0✳ ✹ ❇② ❖❈❆ ♦❜t❛✐♥ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ X =

n<ω Xn✱ s✉❝❤ t❤❛t

[Xn]2 ⊆ K1✱ ❢♦r ❛❧❧ n < ω✳

✺ ❋♦r ❡✈❡r②

❝❤♦♦s❡ ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t ❞❡♥s❡ ✐♥ ✇✐t❤ r❡s♣❡❝t t♦ ✳

✻ ❈❤♦♦s❡ ❛

❢♦r ❡✈❡r② s✉❝❤ t❤❛t ✱ ❛♥❞ ❞❡✜♥❡

✼ ❙❤♦✇ t❤❛t

✐s tr✐✈✐❛❧ ♦♥ ❡✈❡r② ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-37
SLIDE 37

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

Pr♦♦❢ ✭❝♦♥t✐♥✉❡❞✮

✶ ❈♦♥s✐❞❡r t❤❡ t♦♣♦❧♦❣② τ ♦♥ X ♦❜t❛✐♥❡❞ ❜② ✐❞❡♥t✐❢②✐♥❣ a, b

✇✐t❤ a, b, F(a), F(b) ∈ P(ω)4✳ ❚❤❡♥✱ (X, τ) ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡✳

✷ ❙❤♦✇ t❤❛t K0 ✐s ♦♣❡♥ ✐♥ τ s♦ t❤❛t ✇❡ ❝❛♥ ❝❛❧❧ ✉♣♦♥ ❖❈❆✳ ✸ ❚❤❡r❡ ✐s ♥♦ ✉♥❝♦✉♥t❛❜❧❡ Y ⊆ X s✉❝❤ t❤❛t [Y ]2 ⊆ K0✳ ✹ ❇② ❖❈❆ ♦❜t❛✐♥ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ X =

n<ω Xn✱ s✉❝❤ t❤❛t

[Xn]2 ⊆ K1✱ ❢♦r ❛❧❧ n < ω✳

✺ ❋♦r ❡✈❡r② n < ω ❝❤♦♦s❡ ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t Dn ⊆ Xn ❞❡♥s❡ ✐♥

Xn ✇✐t❤ r❡s♣❡❝t t♦ τ✳

✻ ❈❤♦♦s❡ ❛

❢♦r ❡✈❡r② s✉❝❤ t❤❛t ✱ ❛♥❞ ❞❡✜♥❡

✼ ❙❤♦✇ t❤❛t

✐s tr✐✈✐❛❧ ♦♥ ❡✈❡r② ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-38
SLIDE 38

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

Pr♦♦❢ ✭❝♦♥t✐♥✉❡❞✮

✶ ❈♦♥s✐❞❡r t❤❡ t♦♣♦❧♦❣② τ ♦♥ X ♦❜t❛✐♥❡❞ ❜② ✐❞❡♥t✐❢②✐♥❣ a, b

✇✐t❤ a, b, F(a), F(b) ∈ P(ω)4✳ ❚❤❡♥✱ (X, τ) ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡✳

✷ ❙❤♦✇ t❤❛t K0 ✐s ♦♣❡♥ ✐♥ τ s♦ t❤❛t ✇❡ ❝❛♥ ❝❛❧❧ ✉♣♦♥ ❖❈❆✳ ✸ ❚❤❡r❡ ✐s ♥♦ ✉♥❝♦✉♥t❛❜❧❡ Y ⊆ X s✉❝❤ t❤❛t [Y ]2 ⊆ K0✳ ✹ ❇② ❖❈❆ ♦❜t❛✐♥ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ X =

n<ω Xn✱ s✉❝❤ t❤❛t

[Xn]2 ⊆ K1✱ ❢♦r ❛❧❧ n < ω✳

✺ ❋♦r ❡✈❡r② n < ω ❝❤♦♦s❡ ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t Dn ⊆ Xn ❞❡♥s❡ ✐♥

Xn ✇✐t❤ r❡s♣❡❝t t♦ τ✳

✻ ❈❤♦♦s❡ ❛ σ(a) ∈ A ❢♦r ❡✈❡r② a, b ∈ X s✉❝❤ t❤❛t

b ⊆ a ⊆ σ(a)✱ ❛♥❞ ❞❡✜♥❡ B = {σ(a) : (∃n < ω)(a, b ∈ Dn)}

✼ ❙❤♦✇ t❤❛t

✐s tr✐✈✐❛❧ ♦♥ ❡✈❡r② ✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

slide-39
SLIDE 39

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❖❈❆ ✐♥ ❛❝t✐♦♥

Pr♦♦❢ ✭❝♦♥t✐♥✉❡❞✮

✶ ❈♦♥s✐❞❡r t❤❡ t♦♣♦❧♦❣② τ ♦♥ X ♦❜t❛✐♥❡❞ ❜② ✐❞❡♥t✐❢②✐♥❣ a, b

✇✐t❤ a, b, F(a), F(b) ∈ P(ω)4✳ ❚❤❡♥✱ (X, τ) ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡✳

✷ ❙❤♦✇ t❤❛t K0 ✐s ♦♣❡♥ ✐♥ τ s♦ t❤❛t ✇❡ ❝❛♥ ❝❛❧❧ ✉♣♦♥ ❖❈❆✳ ✸ ❚❤❡r❡ ✐s ♥♦ ✉♥❝♦✉♥t❛❜❧❡ Y ⊆ X s✉❝❤ t❤❛t [Y ]2 ⊆ K0✳ ✹ ❇② ❖❈❆ ♦❜t❛✐♥ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ X =

n<ω Xn✱ s✉❝❤ t❤❛t

[Xn]2 ⊆ K1✱ ❢♦r ❛❧❧ n < ω✳

✺ ❋♦r ❡✈❡r② n < ω ❝❤♦♦s❡ ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t Dn ⊆ Xn ❞❡♥s❡ ✐♥

Xn ✇✐t❤ r❡s♣❡❝t t♦ τ✳

✻ ❈❤♦♦s❡ ❛ σ(a) ∈ A ❢♦r ❡✈❡r② a, b ∈ X s✉❝❤ t❤❛t

b ⊆ a ⊆ σ(a)✱ ❛♥❞ ❞❡✜♥❡ B = {σ(a) : (∃n < ω)(a, b ∈ Dn)}

✼ ❙❤♦✇ t❤❛t ϕ ✐s tr✐✈✐❛❧ ♦♥ ❡✈❡r② c ∈ A \ B✳ ❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆

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SLIDE 40

■♥tr♦❞✉❝t✐♦♥ βω ✉♥❞❡r t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s βω ✉♥❞❡r ❖❈❆ Pr❡❧✐♠✐♥❛r✐❡s ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❆✉t♦❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ βω

❘❡❢❡r❡♥❝❡s ■

❆❧❛♥ ❉❖❲ ❡t ❑❧❛❛s P✐❡t❡r ❍❆❘❚✳ ❚❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ❞♦❡s ♥♦t ❛❧✇❛②s ❡♠❜❡❞✳ ❋✉♥❞❛♠❡♥t❛ ▼❛t❤❡♠❛t✐❝❛❡✱ ❱♦❧✳ ✶✻✸ ✭✷✵✵✵✮✱ ♣♣✳ ✶✻✸✲✶✼✻✳ ❙t❡✈♦ ❚❖❉❖❘❷❊❱■❶✳ P❛rt✐t✐♦♥ ♣r♦❜❧❡♠s ✐♥ t♦♣♦❧♦❣②✳ ❈♦♥t❡♠♣♦r❛r② ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✸✹✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✶✾✽✾✳ ❏❛♥ ❱❆◆ ▼■▲▲✳ ■♥tr♦❞✉❝t✐♦♥ t♦ βω✱ ✐♥ ❑✳ ❑✉♥❡♥ ❛♥❞ ❏✳ ❊✳ ❱❛✉❣❤❛♥✱ ❡❞s✳✱ ❍❛♥❞❜♦♦❦ ♦❢ ❙❡t✲❚❤❡♦r❡t✐❝ ❚♦♣♦❧♦❣② ✭◆♦rt❤✲❍♦❧❧❛♥❞✱ ❆♠st❡r❞❛♠✱ ✶✾✽✹✮ ✺✵✸✲✺✻✼✳ ❇♦❜❛♥ ❱❊▲■❷❑❖❱■❶✳ ❖❈❆ ❛♥❞ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ P(ω)/fin✳ ❚♦♣♦❧♦❣② ❛♥❞ ✐s ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✳ ✹✾ ✭✶✾✾✸✮✱ ♣♣✳ ✶✲✶✸✳

❈r✐stó❜❛❧ ❘♦❞rí❣✉❡③ P♦rr❛s βω ❛♥❞ ❖❈❆