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qtt rtr r - - PowerPoint PPT Presentation

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qtt rtr r tr sts ts ts rr rtt

slide-1
SLIDE 1

❊q✉✐❝♦♥t✐♥✉✐t② ❝r✐t❡r✐❛ ❢♦r ♠❡tr✐❝✲✈❛❧✉❡❞ s❡ts ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s

▲✳ ❚árr❡❣❛

❉❡♣❛rt❛♠❡♥t ❞❡ ♠❛t❡♠àt✐q✉❡s✱ ❯♥✐✈❡rs✐t❛t ❏❛✉♠❡ ■✱ ❈❛st❡❧❧ó✱ ❙♣❛✐♥

✹t❤ ❲♦r❦s❤♦♣ ♦♥ ❚♦♣♦❧♦❣✐❝❛❧ ●r♦✉♣s ❉❡❝❡♠❜❡r ✸✲✹✱ ✷✵✶✺✱ ▼❛❞r✐❞✱ ❙♣❛✐♥

❏♦✐♥t ✇♦r❦ ✇✐t❤ ▼✳ ❱✳ ❋❡rr❡r ❛♥❞ ❙✳ ❍❡r♥á♥❞❡③

slide-2
SLIDE 2

■♥❞❡①

✶

■♥tr♦❞✉❝t✐♦♥

✷

▼❛✐♥ ❘❡s✉❧ts

✸

❇❛s✐❝ ❘❡s✉❧ts

✹

❆♣♣❧✐❝❛t✐♦♥s

slide-3
SLIDE 3

■♥tr♦❞✉❝t✐♦♥

■♥❞❡①

✶

■♥tr♦❞✉❝t✐♦♥

✷

▼❛✐♥ ❘❡s✉❧ts

✸

❇❛s✐❝ ❘❡s✉❧ts

✹

❆♣♣❧✐❝❛t✐♦♥s

✶ ✴ ✷✽

slide-4
SLIDE 4

■♥tr♦❞✉❝t✐♦♥

▲❡t X ❛♥❞ (M, d) ❜❡ ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❛♥❞ ❛ ♠❡tr✐❝ s♣❛❝❡ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ ❧❡t G ⊆ C(X, M)✳ ❉❡✜♥✐t✐♦♥

✶ ❆ ♣♦✐♥t x ∈ X ✐s ❛♥ ❡q✉✐❝♦♥t✐♥✉✐t② ♣♦✐♥t ♦❢ G ✇❤❡♥ ❢♦r ❡✈❡r② ǫ > ✵

t❤❡r❡ ✐s ❛ ♥❡✐❣❤❜♦r❤♦♦❞ U ♦❢ x s✉❝❤ t❤❛t diam(g(U)) < ǫ ❢♦r ❛❧❧ g ∈ G✳

✷ G ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ✭❆❊✮ ✇❤❡♥ t❤❡ s✉❜s❡t ♦❢ ❡q✉✐❝♦♥t✐♥✉✐t②

♣♦✐♥ts ♦❢ G ✐s ❞❡♥s❡ ✐♥ X✳

✸ G ✐s ❤❡r❡❞✐t❛r✐❧② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ✭❍❆❊✮ ✇❤❡♥ G|A ✐s

❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ❢♦r ❡✈❡r② s✉❜s❡t A ♦❢ X✳

✶ ✴ ✷✽

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SLIDE 5

■♥tr♦❞✉❝t✐♦♥

▲❡♠♠❛ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ♣r♦♣❡rt✐❡s✿ ✭❛✮ G ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❜✮ ❋♦r ❡✈❡r② ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t U ♦❢ X ❛♥❞ ǫ > ✵✱ t❤❡r❡ ❡①✐sts ❛ ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t V ⊆ U s✉❝❤ t❤❛t diam(g(V )) < ǫ ❢♦r ❛❧❧ g ∈ G✳ ❚❤❡♥ ✭❛✮ ✐♠♣❧✐❡s ✭❜✮✳

✷ ✴ ✷✽

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SLIDE 6

■♥tr♦❞✉❝t✐♦♥

■❢ ❳ ✐s ❛ ❇❛✐r❡ s♣❛❝❡✱ t❤❡♥ ✭❛✮ ❛♥❞ ✭❜✮ ❛r❡ ❡q✉✐✈❛❧❡♥t✳ Pr♦♦❢ ✿ ✲ ❈♦♥s✐❞❡r t❤❡ ♦♣❡♥ s❡t Oǫ := {U ⊆ X : U ✐s ❛ ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t ∧ diam(g(U)) < ǫ ∀g ∈ G}✳

✷ ✴ ✷✽

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SLIDE 7

■♥tr♦❞✉❝t✐♦♥

■❢ ❳ ✐s ❛ ❇❛✐r❡ s♣❛❝❡✱ t❤❡♥ ✭❛✮ ❛♥❞ ✭❜✮ ❛r❡ ❡q✉✐✈❛❧❡♥t✳ Pr♦♦❢ ✿ ✲ ❈♦♥s✐❞❡r t❤❡ ♦♣❡♥ s❡t Oǫ := {U ⊆ X : U ✐s ❛ ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t ∧ diam(g(U)) < ǫ ∀g ∈ G}✳ ✲ ❇② ✭❜✮✱ Oǫ = ∅ ❛♥❞ ❞❡♥s❡ ✐♥ X✳

✷ ✴ ✷✽

slide-8
SLIDE 8

■♥tr♦❞✉❝t✐♦♥

■❢ ❳ ✐s ❛ ❇❛✐r❡ s♣❛❝❡✱ t❤❡♥ ✭❛✮ ❛♥❞ ✭❜✮ ❛r❡ ❡q✉✐✈❛❧❡♥t✳ Pr♦♦❢ ✿ ✲ ❈♦♥s✐❞❡r t❤❡ ♦♣❡♥ s❡t Oǫ := {U ⊆ X : U ✐s ❛ ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t ∧ diam(g(U)) < ǫ ∀g ∈ G}✳ ✲ ❇② ✭❜✮✱ Oǫ = ∅ ❛♥❞ ❞❡♥s❡ ✐♥ X✳ ✲ ❙✐♥❝❡ X ✐s ❇❛✐r❡✱ t❛❦✐♥❣ W :=

n∈ω

O ✶

n = ∅✱ ✇❡ ♦❜t❛✐♥ ❛ ❞❡♥s❡ Gδ s✉❜s❡t

✇❤✐❝❤ ✐s t❤❡ s✉❜s❡t ♦❢ ❡q✉✐❝♦♥t✐♥✉✐t② ♣♦✐♥ts ♦❢ G✳

✷ ✴ ✷✽

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SLIDE 9

■♥tr♦❞✉❝t✐♦♥

❊①❛♠♣❧❡ ▲❡t X = ✷ω ❜❡ t❤❡ ❈❛♥t♦r s♣❛❝❡ ❛♥❞ ❧❡t G = {πn}n∈ω ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣r♦❥❡❝t✐♦♥s ♦❢ X ♦♥t♦ {✵, ✶}✳ ❚❤❡♥ G ✐s ♥♦t ❆❊✳

✸ ✴ ✷✽

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SLIDE 10

■♥tr♦❞✉❝t✐♦♥

❊①❛♠♣❧❡ ▲❡t X = R ❛♥❞ ❧❡t G = {❛r❝t❛♥(nx)}n∈ω ❚❤❡♥ G ✐s ❍❆❊✳

✸ ✴ ✷✽

slide-11
SLIDE 11

■♥tr♦❞✉❝t✐♦♥

❘❡❝❛❧❧ t❤❛t ❛ ❞②♥❛♠✐❝❛❧ s②st❡♠✱ ♦r ❛ G✲s♣❛❝❡✱ ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡ X ♦♥ ✇❤✐❝❤ ❛ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣ G ❛❝ts ❝♦♥t✐♥✉♦✉s❧②✳ ❋♦r ❡❛❝❤ g ∈ G ✇❡ ❤❛✈❡ t❤❡ s❡❧❢✲❤♦♠❡♦♠♦r♣❤✐s♠ x → gx ♦❢ X t❤❛t ✇❡ ❝❛❧❧ g✲tr❛♥s❧❛t✐♦♥✳ ❊①❛♠♣❧❡ ✭●❧❛s♥❡r ❛♥❞ ▼❡❣r❡❧✐s❤✈✐❧✐✮ ❚❤❡r❡ ❛r❡ ❞②♥❛♠✐❝❛❧ s②st❡♠s ❆❊ ✇❤✐❝❤ ❛r❡ ♥♦t ❍❆❊✳

✸ ✴ ✷✽

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SLIDE 12

▼❛✐♥ ❘❡s✉❧ts

■♥❞❡①

✶

■♥tr♦❞✉❝t✐♦♥

✷

▼❛✐♥ ❘❡s✉❧ts

✸

❇❛s✐❝ ❘❡s✉❧ts

✹

❆♣♣❧✐❝❛t✐♦♥s

✹ ✴ ✷✽

slide-13
SLIDE 13

▼❛✐♥ ❘❡s✉❧ts

❚❤❡♦r❡♠ ❆ ▲❡t X ❛♥❞ (M, d) ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ s♣❛❝❡ ❛♥❞ ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ ❧❡t G ⊆ C(X, M) s✉❝❤ t❤❛t G

MX

✐s ❝♦♠♣❛❝t✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s✿ ✭❛✮ G ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❜✮ ❚❤❡r❡ ❡①✐sts ❛ ❞❡♥s❡ ❇❛✐r❡ s✉❜s❡t F ⊆ X s✉❝❤ t❤❛t (G

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✳ ✭❝✮ ❚❤❡r❡ ❡①✐sts ❛ ❞❡♥s❡ Gδ s✉❜s❡t F ⊆ X s✉❝❤ t❤❛t (F, tp(G

MX

)) ✐s ▲✐♥❞❡❧ö❢✳ ❚❤❡♥ (b) ⇒ (c) ⇒ (a)✳ ■❢ X ✐s ❛❧s♦ ❛ ❤❡r❡❞✐t❛r✐❧② ▲✐♥❞❡❧ö❢ s♣❛❝❡✱ t❤❡♥ ❛❧❧ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✳

✹ ✴ ✷✽

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SLIDE 14

▼❛✐♥ ❘❡s✉❧ts

❚❤❡♦r❡♠ ❇ ▲❡t X ❛♥❞ (M, d) ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ s♣❛❝❡ ❛♥❞ ❛ ♠❡tr✐❝ s♣❛❝❡✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ ❧❡t G ⊆ C(X, M) s✉❝❤ t❤❛t G

MX

✐s ❝♦♠♣❛❝t✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✭❛✮ G ✐s ❤❡r❡❞✐t❛r② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❜✮ L ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ♦♥ F✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ✭❝✮ (L

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ✭❞✮ (F, tp(L

MX

)) ✐s ▲✐♥❞❡❧ö❢✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳

✺ ✴ ✷✽

slide-15
SLIDE 15

❇❛s✐❝ ❘❡s✉❧ts

■♥❞❡①

✶

■♥tr♦❞✉❝t✐♦♥

✷

▼❛✐♥ ❘❡s✉❧ts

✸

❇❛s✐❝ ❘❡s✉❧ts

✹

❆♣♣❧✐❝❛t✐♦♥s

✻ ✴ ✷✽

slide-16
SLIDE 16

❇❛s✐❝ ❘❡s✉❧ts

❉❡✜♥✐t✐♦♥ X ✐s s❛✐❞ t♦ ❜❡ ❤❡♠✐❝♦♠♣❛❝t ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❛❝ts s❡ts {Xn}n∈ω s✉❝❤ t❤❛t X =

n∈ω

Xn ❛♥❞ ❢♦r ❡✈❡r② ❝♦♠♣❛❝t s✉❜s❡t C ♦❢ X t❤❡r❡ ✐s n ∈ ω s✉❝❤ t❤❛t C ⊆ Xn✳

✻ ✴ ✷✽

slide-17
SLIDE 17

❇❛s✐❝ ❘❡s✉❧ts

▲❡♠♠❛ ✶ ▲❡t X ❛♥❞ (M, d) ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ s♣❛❝❡ ❛♥❞ ❛ ❤❡♠✐❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ ❧❡t G ❜❡ ❛ s✉❜s❡t ♦❢ C(X, M) s✉❝❤ t❤❛t G

MX

✐s ❝♦♠♣❛❝t✳ ■❢ ● ✐s ♥♦t ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✱ t❤❡♥ ❢♦r ❡✈❡r② Gδ ❛♥❞ ❞❡♥s❡ s✉❜s❡t F ♦❢ X t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t L ✐♥ G✱ ❛ ❝♦♠♣❛❝t s❡♣❛r❛❜❧❡ s✉❜s❡t KF ⊆ F✱ ❛ ❝♦♠♣❛❝t s✉❜s❡t N ⊆ M ❛♥❞ ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ s✉r❥❡❝t✐✈❡ ♠❛♣ Ψ ♦❢ KF ♦♥t♦ t❤❡ ❈❛♥t♦r s❡t ✷ω s✉❝❤ t❤❛t ✐❢ t❤❡ ♠❛♣s l∗ ❛r❡ ❞❡✜♥❡❞ t♦ ♠❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠ ❝♦♠♠✉t❛t✐✈❡ KF

Ψ

  • l|KF
  • ✷ω

l∗

  • N

t❤❡♥ t❤❡ s✉❜s❡t ▲∗ := {❧ ∗ : ❧ ∈ ▲} ⊆ C(✷ω, N) ✐s ♥♦t ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ♦♥ ✷ω✳

✼ ✴ ✷✽

slide-18
SLIDE 18

❇❛s✐❝ ❘❡s✉❧ts

❘❡♠❛r❦ ▲❡t X ❜❡ ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ (M, d) ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ G ❜❡ ❛ s✉❜s❡t ♦❢ C(X, M)✳ ❙❡t K := {α : M → [−✶, ✶] : |α(m✶) − α(m✷)| ≤ d(m✶, m✷), ∀m✶, m✷ ∈ M}. K ✐s ❛ ❝♦♠♣❛❝t s✉❜s♣❛❝❡ ♦❢ [−✶, ✶]M✳

✽ ✴ ✷✽

slide-19
SLIDE 19

❇❛s✐❝ ❘❡s✉❧ts

❘❡♠❛r❦ ❈♦♥s✐❞❡r t❤❡ ❡✈❛❧✉❛t✐♦♥ ♠❛♣ ϕ : X × G − → M (x, g) − → g(x) ✇❤✐❝❤ ✐s ❝❧❡❛r❧② s❡♣❛r❛t❡❧② ❝♦♥t✐♥✉♦✉s✳

✾ ✴ ✷✽

slide-20
SLIDE 20

❇❛s✐❝ ❘❡s✉❧ts

❘❡♠❛r❦ ❈♦♥s✐❞❡r t❤❡ ❡✈❛❧✉❛t✐♦♥ ♠❛♣ ϕ : X × G − → M (x, g) − → g(x) ✇❤✐❝❤ ✐s ❝❧❡❛r❧② s❡♣❛r❛t❡❧② ❝♦♥t✐♥✉♦✉s✳ ❚❤❡ ♠❛♣ ϕ ❤❛s ❛ss♦❝✐❛t❡❞ ❛ s❡♣❛r❛t❡❧② ❝♦♥t✐♥✉♦✉s ♠❛♣ ϕ∗ : X × (G × K) − → [−✶, ✶] (x, (g, α)) − → α(g(x))

✾ ✴ ✷✽

slide-21
SLIDE 21

❇❛s✐❝ ❘❡s✉❧ts

❘❡♠❛r❦ ❙❡t ν : G

MX

× K − → [−✶, ✶]X (h, α) − → α ◦ h ❙✐♥❝❡ G ⊆ C(X, M)✱ ✇❡ ❤❛✈❡ t❤❛t ν(G × K) ⊆ C(X, [−✶, ✶])✳ ¯ d : M × M → R ❞❡✜♥❡❞ ❜② ¯ d(m✶, m✷) := ♠✐♥{d(m✶, m✷), ✶} ∀m✶, m✷ ∈ M

✶✵ ✴ ✷✽

slide-22
SLIDE 22

❇❛s✐❝ ❘❡s✉❧ts

▲❡♠♠❛ ✷ ▲❡t X ❛♥❞ (M, d) ❜❡ ❛ t♦♣♦❧♦❣✐❝❛❧ ❛♥❞ ❛ ♠❡tr✐❝ s♣❛❝❡✱ r❡s♣❡❝t✐✈❡❧②✳ ■❢ G ✐s ❛ s✉❜s❡t ♦❢ C(X, M)✱ t❤❡♥ G ✐s ❡q✉✐❝♦♥t✐♥✉♦✉s ❛t ❛ ♣♦✐♥t x✵ ∈ X ✐❢ ❛♥❞ ♦♥❧② ✐❢ ν(G × K) ✐s ❡q✉✐❝♦♥t✐♥✉♦✉s ❛t ✐t✳

✶✶ ✴ ✷✽

slide-23
SLIDE 23

❇❛s✐❝ ❘❡s✉❧ts

▲❡♠♠❛ ✷ ▲❡t X ❛♥❞ (M, d) ❜❡ ❛ t♦♣♦❧♦❣✐❝❛❧ ❛♥❞ ❛ ♠❡tr✐❝ s♣❛❝❡✱ r❡s♣❡❝t✐✈❡❧②✳ ■❢ G ✐s ❛ s✉❜s❡t ♦❢ C(X, M)✱ t❤❡♥ G ✐s ❡q✉✐❝♦♥t✐♥✉♦✉s ❛t ❛ ♣♦✐♥t x✵ ∈ X ✐❢ ❛♥❞ ♦♥❧② ✐❢ ν(G × K) ✐s ❡q✉✐❝♦♥t✐♥✉♦✉s ❛t ✐t✳ Pr♦♦❢ ✿ (⇒) ●✐✈❡♥ ǫ > ✵✱ t❤❡r❡ ✐s ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦✉❤♦♦❞ U ♦❢ x✵ s✉❝❤ t❤❛t d(g(x✵), g(x)) < ǫ ❢♦r ❛❧❧ x ∈ U ❛♥❞ g ∈ G✳

✶✶ ✴ ✷✽

slide-24
SLIDE 24

❇❛s✐❝ ❘❡s✉❧ts

▲❡♠♠❛ ✷ ▲❡t X ❛♥❞ (M, d) ❜❡ ❛ t♦♣♦❧♦❣✐❝❛❧ ❛♥❞ ❛ ♠❡tr✐❝ s♣❛❝❡✱ r❡s♣❡❝t✐✈❡❧②✳ ■❢ G ✐s ❛ s✉❜s❡t ♦❢ C(X, M)✱ t❤❡♥ G ✐s ❡q✉✐❝♦♥t✐♥✉♦✉s ❛t ❛ ♣♦✐♥t x✵ ∈ X ✐❢ ❛♥❞ ♦♥❧② ✐❢ ν(G × K) ✐s ❡q✉✐❝♦♥t✐♥✉♦✉s ❛t ✐t✳ Pr♦♦❢ ✿ (⇒) ●✐✈❡♥ ǫ > ✵✱ t❤❡r❡ ✐s ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦✉❤♦♦❞ U ♦❢ x✵ s✉❝❤ t❤❛t d(g(x✵), g(x)) < ǫ ❢♦r ❛❧❧ x ∈ U ❛♥❞ g ∈ G✳ ▲❡t α ∈ K✱ x ∈ U ❛♥❞ g ∈ G✱ t❤❡♥ ✇❡ ❤❛✈❡ |ν(g, α)(x✵) − ν(g, α)(x)| = |α(g(x✵)) − α(g(x))| ≤ d(g(x✵), g(x)) < ǫ.

✶✶ ✴ ✷✽

slide-25
SLIDE 25

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ✿ (⇐) ●✐✈❡♥ ǫ > ✵✱ t❤❡r❡ ✐s ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦✉❤♦♦❞ U ♦❢ x✵ s✉❝❤ t❤❛t |ν(g, α)(x✵) − ν(g, α)(x)| < ǫ ❢♦r ❛❧❧ x ∈ U✱ g ∈ G ❛♥❞ α ∈ K✳

✶✶ ✴ ✷✽

slide-26
SLIDE 26

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ✿ (⇐) ●✐✈❡♥ ǫ > ✵✱ t❤❡r❡ ✐s ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦✉❤♦♦❞ U ♦❢ x✵ s✉❝❤ t❤❛t |ν(g, α)(x✵) − ν(g, α)(x)| < ǫ ❢♦r ❛❧❧ x ∈ U✱ g ∈ G ❛♥❞ α ∈ K✳ ❙❡t α✵ ∈ [−✶, ✶]M ❞❡✜♥❡❞ ❜② α✵(m) ❞❡❢ = d(m, g(x✵)) ❢♦r ❛❧❧ m ∈ M✳ ■t ✐s ❡❛s② t♦ ❝❤❡❝❦ t❤❛t α✵ ∈ K✳

✶✶ ✴ ✷✽

slide-27
SLIDE 27

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ✿ (⇐) ●✐✈❡♥ ǫ > ✵✱ t❤❡r❡ ✐s ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦✉❤♦♦❞ U ♦❢ x✵ s✉❝❤ t❤❛t |ν(g, α)(x✵) − ν(g, α)(x)| < ǫ ❢♦r ❛❧❧ x ∈ U✱ g ∈ G ❛♥❞ α ∈ K✳ ❙❡t α✵ ∈ [−✶, ✶]M ❞❡✜♥❡❞ ❜② α✵(m) ❞❡❢ = d(m, g(x✵)) ❢♦r ❛❧❧ m ∈ M✳ ■t ✐s ❡❛s② t♦ ❝❤❡❝❦ t❤❛t α✵ ∈ K✳ ■t s✉✣❝❡s t♦ ♦❜s❡r✈❡ t❤❛t |α✵(g(x✵)) − α✵(g(x))| = d(g(x), g(x✵)) ❢♦r ❛❧❧ x ∈ U ❛♥❞ g ∈ G✳

✶✶ ✴ ✷✽

slide-28
SLIDE 28

❇❛s✐❝ ❘❡s✉❧ts

❈♦r♦❧❧❛r② ✶ ▲❡t X ❛♥❞ (M, d) ❜❡ ❛ t♦♣♦❧♦❣✐❝❛❧ ❛♥❞ ❛ ♠❡tr✐❝ s♣❛❝❡✱ r❡s♣❡❝t✐✈❡❧②✳ ■❢ G ✐s ❛ s✉❜s❡t ♦❢ C(X, M)✱ t❤❡♥ G ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ν(G × K) ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳

✶✷ ✴ ✷✽

slide-29
SLIDE 29

❇❛s✐❝ ❘❡s✉❧ts

❚❤❡♦r❡♠ ✭❈❛s❝❛❧❡s✱ ◆❛♠✐♦❦❛ ❛♥❞ ❱❡r❛✮ ▲❡t X ❜❡ ❛ ❝♦♠♣❛❝t s♣❛❝❡✱ (M, d) ❜❡ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❧❡t G ❜❡ ❛ s✉❜s❡t ♦❢ C(X, M)✳ ■❢ (X, tp(G

MX

)) ✐s ▲✐♥❞❡❧ö❢✱ t❤❡♥ G ✐s ❤❡r❡❞✐t❛r✐❧② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳

✶✸ ✴ ✷✽

slide-30
SLIDE 30

❇❛s✐❝ ❘❡s✉❧ts

❚❤❡♦r❡♠ ✶ ▲❡t X ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ s♣❛❝❡✱ (M, d) ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❧❡t G ❜❡ ❛ s✉❜s❡t ♦❢ C(X, M) s✉❝❤ t❤❛t G

MX

✐s ❝♦♠♣❛❝t✳ ■❢ t❤❡r❡ ❡①✐sts ❛ ❛ ❞❡♥s❡ Gδ s✉❜s❡t F ⊆ X s✉❝❤ t❤❛t (F, tp(G

MX

)) ✐s ▲✐♥❞❡❧ö❢✱ t❤❡♥ G ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳

✶✹ ✴ ✷✽

slide-31
SLIDE 31

❇❛s✐❝ ❘❡s✉❧ts

❚❤❡♦r❡♠ ✭●❧❛s♥❡r✱ ▼❡❣r❡❧✐s❤✈✐❧✐ ❛♥❞ ❯s♣❡♥s❦✐❥✮ ▲❡t X ❜❡ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✱ (M, d) ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❧❡t G ⊆ C(X, M) ❛♥❞ H = G

MX

✳ ■❢ H ✐s ❝♦♠♣❛❝t ❤❡r❡❞✐t❛r② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✱ t❤❡♥ H ✐s ♠❡tr✐③❛❜❧❡✳ ▲❡♠♠❛ ✭●❧❛s♥❡r ❛♥❞ ▼❡❣r❡❧✐s❤✈✐❧✐✮ ▲❡t X✱ Y ❛♥❞ (M, d) ❜❡ t✇♦ ❛r❜✐tr❛r② ❝♦♠♣❛❝t s♣❛❝❡s ❛♥❞ ❛ ♠❡tr✐❝ s♣❛❝❡✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ ❧❡t G ❜❡ ❛ s✉❜s❡t ♦❢ C(Y , M)✳ ❙✉♣♣♦s❡ t❤❛t p : X → Y ✐s ❛ ❝♦♥t✐♥✉♦✉s ♦♥t♦ ♠❛♣✳ X

p

  • G◦p∋g◦p
  • Y

g∈G

  • M

❚❤❡♥ G ◦ p := {g ◦ p : g ∈ G} ⊆ C(X, M) ✐s ❍❆❊ ✐❢ ❛♥❞ ♦♥❧② ✐❢ G ✐s ❍❆❊✳

✶✺ ✴ ✷✽

slide-32
SLIDE 32

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♣♦s✐t✐♦♥ ✶ ▲❡t X ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ s♣❛❝❡✱ (M, d) ❜❡ ❛ ❤❡♠✐❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ G ⊆ C(X, M) s✉❝❤ t❤❛t G

MX

✐s ❝♦♠♣❛❝t✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✭❛✮ G ✐s ❤❡r❡❞✐t❛r② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❜✮ L ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ♦♥ F✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳

✶✻ ✴ ✷✽

slide-33
SLIDE 33

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ✿ (b) ⇒ (a) ❆ss✉♠❡ t❤❛t ✭❛✮ ❞♦❡s ♥♦t ❤♦❧❞ ⇒ ∃A ⊆ X s✉❝❤ t❤❛t G|A ✐s ♥♦t ❆❊✳

✶✻ ✴ ✷✽

slide-34
SLIDE 34

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ✿ (b) ⇒ (a) ❆ss✉♠❡ t❤❛t ✭❛✮ ❞♦❡s ♥♦t ❤♦❧❞ ⇒ ∃A ⊆ X s✉❝❤ t❤❛t G|A ✐s ♥♦t ❆❊✳ ❇② ▲❡♠♠❛ ✶ t❤❡r❡ ❡①✐sts ❛ ❝♦♠♣❛❝t ❛♥❞ s❡♣❛r❛❜❧❡ s✉❜s❡t F ♦❢ X ✐♥❝❧✉❞❡❞ ✐♥ A✱ ❛♥ ♦♥t♦ ❛♥❞ ❝♦♥t✐♥✉♦✉s ♠❛♣ Ψ : F → ✷ω✱ ❛♥❞ ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t L ♦❢ G s✉❝❤ t❤❛t t❤❡ s✉❜s❡t L∗ ⊆ C(✷ω, M) ❞❡✜♥❡❞ ❜② l∗(Ψ(x)) = l(x) ❢♦r ❛❧❧ x ∈ F ✐s ♥♦t ❆❊✳

✶✻ ✴ ✷✽

slide-35
SLIDE 35

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ✿ (b) ⇒ (a) ❆ss✉♠❡ t❤❛t ✭❛✮ ❞♦❡s ♥♦t ❤♦❧❞ ⇒ ∃A ⊆ X s✉❝❤ t❤❛t G|A ✐s ♥♦t ❆❊✳ ❇② ▲❡♠♠❛ ✶ t❤❡r❡ ❡①✐sts ❛ ❝♦♠♣❛❝t ❛♥❞ s❡♣❛r❛❜❧❡ s✉❜s❡t F ♦❢ X ✐♥❝❧✉❞❡❞ ✐♥ A✱ ❛♥ ♦♥t♦ ❛♥❞ ❝♦♥t✐♥✉♦✉s ♠❛♣ Ψ : F → ✷ω✱ ❛♥❞ ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t L ♦❢ G s✉❝❤ t❤❛t t❤❡ s✉❜s❡t L∗ ⊆ C(✷ω, M) ❞❡✜♥❡❞ ❜② l∗(Ψ(x)) = l(x) ❢♦r ❛❧❧ x ∈ F ✐s ♥♦t ❆❊✳ ❇② ▲❡♠♠❛ ●▼ L ✐s ♥♦t ❆❊ ♦♥ ❋✳ ❈♦♥tr❛❞✐❝t✐♦♥

✶✻ ✴ ✷✽

slide-36
SLIDE 36

❇❛s✐❝ ❘❡s✉❧ts

❈♦r♦❧❧❛r② ✷ ▲❡t X ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ s♣❛❝❡✱ (M, d) ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ G ⊆ C(X, M) s✉❝❤ t❤❛t G

MX

✐s ❝♦♠♣❛❝t✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✭❛✮ G ✐s ❤❡r❡❞✐t❛r② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❜✮ L ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ♦♥ F✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳

✶✼ ✴ ✷✽

slide-37
SLIDE 37

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

✭❛✮ G ✐s ❤❡r❡❞✐t❛r② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❜✮ L ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ♦♥ F✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ (a) ⇔ (b) ❈♦r♦❧❧❛r② ✷✳

✶✽ ✴ ✷✽

slide-38
SLIDE 38

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

✭❛✮ G ✐s ❤❡r❡❞✐t❛r② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❝✮ (L

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ (a) ⇒ (c) ▲❡t L ∈ [G]≤ω ❛♥❞ ❧❡t F ❜❡ ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳

✶✾ ✴ ✷✽

slide-39
SLIDE 39

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

✭❛✮ G ✐s ❤❡r❡❞✐t❛r② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❝✮ (L

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ (a) ⇒ (c) ▲❡t L ∈ [G]≤ω ❛♥❞ ❧❡t F ❜❡ ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ x ∼ y ✐❢ ❛♥❞ ♦♥❧② ✐❢ l(x) = l(y) ❢♦r ❛❧❧ l ∈ L✳

✶✾ ✴ ✷✽

slide-40
SLIDE 40

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

✭❛✮ G ✐s ❤❡r❡❞✐t❛r② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❝✮ (L

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ (a) ⇒ (c) ▲❡t L ∈ [G]≤ω ❛♥❞ ❧❡t F ❜❡ ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ x ∼ y ✐❢ ❛♥❞ ♦♥❧② ✐❢ l(x) = l(y) ❢♦r ❛❧❧ l ∈ L✳

  • F = F/∼ ✐s t❤❡ ❝♦♠♣❛❝t q✉♦t✐❡♥t s♣❛❝❡ ❛♥❞ p : F →

F ✐s t❤❡ q✉♦t✐❡♥t ♠❛♣✳

✶✾ ✴ ✷✽

slide-41
SLIDE 41

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

✭❛✮ G ✐s ❤❡r❡❞✐t❛r② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❝✮ (L

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ (a) ⇒ (c) ▲❡t L ∈ [G]≤ω ❛♥❞ ❧❡t F ❜❡ ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ x ∼ y ✐❢ ❛♥❞ ♦♥❧② ✐❢ l(x) = l(y) ❢♦r ❛❧❧ l ∈ L✳

  • F = F/∼ ✐s t❤❡ ❝♦♠♣❛❝t q✉♦t✐❡♥t s♣❛❝❡ ❛♥❞ p : F →

F ✐s t❤❡ q✉♦t✐❡♥t ♠❛♣✳ ❊❛❝❤ l ∈ L ❤❛s ❛ss♦❝✐❛t❡❞ ❛ ♠❛♣ ˜ l ∈ C( F, M) ❞❡✜♥❡❞ ❛s ˜ l(˜ x)

❞❡❢

= l(x) ❢♦r ❛♥② x ∈ F ✇✐t❤ p(x) = ˜ x✳

✶✾ ✴ ✷✽

slide-42
SLIDE 42

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

❊❛❝❤ l ∈ L

MF

❤❛s ❛ss♦❝✐❛t❡❞ ❛ ♠❛♣ ˜ l ∈ ˜ L

M

F

s✉❝❤ t❤❛t ˜ l ◦ p = l✳

✶✾ ✴ ✷✽

slide-43
SLIDE 43

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

❊❛❝❤ l ∈ L

MF

❤❛s ❛ss♦❝✐❛t❡❞ ❛ ♠❛♣ ˜ l ∈ ˜ L

M

F

s✉❝❤ t❤❛t ˜ l ◦ p = l✳ ( F, tp(˜ L)) ✐s ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✳

✶✾ ✴ ✷✽

slide-44
SLIDE 44

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

❊❛❝❤ l ∈ L

MF

❤❛s ❛ss♦❝✐❛t❡❞ ❛ ♠❛♣ ˜ l ∈ ˜ L

M

F

s✉❝❤ t❤❛t ˜ l ◦ p = l✳ ( F, tp(˜ L)) ✐s ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✳ L ✐s ❍❆❊ ♦♥ F

GM

= = ⇒ L ✐s ❍❆❊ ♦♥ F✳

GMU

= = = ⇒ ˜ L

M

F

✐s ♠❡tr✐③❛❜❧❡✳

✶✾ ✴ ✷✽

slide-45
SLIDE 45

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

❊❛❝❤ l ∈ L

MF

❤❛s ❛ss♦❝✐❛t❡❞ ❛ ♠❛♣ ˜ l ∈ ˜ L

M

F

s✉❝❤ t❤❛t ˜ l ◦ p = l✳ ( F, tp(˜ L)) ✐s ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✳ L ✐s ❍❆❊ ♦♥ F

GM

= = ⇒ L ✐s ❍❆❊ ♦♥ F✳

GMU

= = = ⇒ ˜ L

M

F

✐s ♠❡tr✐③❛❜❧❡✳ L

MF

✐s ❝❛♥♦♥✐❝❛❧❧② ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ˜ L

M

F

⇒ L

MF

✐s ♠❡tr✐③❛❜❧❡✳

✶✾ ✴ ✷✽

slide-46
SLIDE 46

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

✭❝✮ (L

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ✭❞✮ (F, tp(L

MX

)) ✐s ▲✐♥❞❡❧ö❢✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ (c) ⇒ (d) H := ((L

MX

)|F, tp(F)) ✐s ❝♦♠♣❛❝t ♠❡tr✐❝✳

✷✵ ✴ ✷✽

slide-47
SLIDE 47

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

✭❝✮ (L

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ✭❞✮ (F, tp(L

MX

)) ✐s ▲✐♥❞❡❧ö❢✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ (c) ⇒ (d) H := ((L

MX

)|F, tp(F)) ✐s ❝♦♠♣❛❝t ♠❡tr✐❝✳ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t M ✐s s❡♣❛r❛❜❧❡ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✳

✷✵ ✴ ✷✽

slide-48
SLIDE 48

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

✭❝✮ (L

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ✭❞✮ (F, tp(L

MX

)) ✐s ▲✐♥❞❡❧ö❢✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ (c) ⇒ (d) H := ((L

MX

)|F, tp(F)) ✐s ❝♦♠♣❛❝t ♠❡tr✐❝✳ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t M ✐s s❡♣❛r❛❜❧❡ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✳ ❊✈❡r② ❡❧❡♠❡♥t x ∈ F ❤❛s ❛ss♦❝✐❛t❡❞ ❛♥ ❡❧❡♠❡♥t ˆ x ∈ M

H ❞❡✜♥❡❞ ❜② ˆ

x(h)

❞❡❢

= h(x) ❢♦r ❛❧❧ h ∈ H✳

✷✵ ✴ ✷✽

slide-49
SLIDE 49

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

✭❝✮ (L

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ✭❞✮ (F, tp(L

MX

)) ✐s ▲✐♥❞❡❧ö❢✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ (c) ⇒ (d) H := ((L

MX

)|F, tp(F)) ✐s ❝♦♠♣❛❝t ♠❡tr✐❝✳ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t M ✐s s❡♣❛r❛❜❧❡ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✳ ❊✈❡r② ❡❧❡♠❡♥t x ∈ F ❤❛s ❛ss♦❝✐❛t❡❞ ❛♥ ❡❧❡♠❡♥t ˆ x ∈ M

H ❞❡✜♥❡❞ ❜② ˆ

x(h)

❞❡❢

= h(x) ❢♦r ❛❧❧ h ∈ H✳ ❙❡t F := {ˆ x : x ∈ F}✳

✷✵ ✴ ✷✽

slide-50
SLIDE 50

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

❙✐♥❝❡ H ✐s ❝♦♠♣❛❝t ♠❡tr✐❝ ⇒ (F, t∞(H)) ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡ ⇒ (F, t∞(H)) ✐s ▲✐♥❞❡❧ö❢✳

✷✵ ✴ ✷✽

slide-51
SLIDE 51

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

❙✐♥❝❡ H ✐s ❝♦♠♣❛❝t ♠❡tr✐❝ ⇒ (F, t∞(H)) ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ♠❡tr✐③❛❜❧❡ ⇒ (F, t∞(H)) ✐s ▲✐♥❞❡❧ö❢✳ ⇒ (F, tp(H)) ✐s ▲✐♥❞❡❧ö❢✳

✷✵ ✴ ✷✽

slide-52
SLIDE 52

❇❛s✐❝ ❘❡s✉❧ts

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ❇

✭❜✮ L ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ♦♥ F✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ✭❞✮ (F, tp(L

MX

)) ✐s ▲✐♥❞❡❧ö❢✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ (d) ⇒ (b) ❆♣♣❧② ❚❤❡♦r❡♠ ✶✳

✷✶ ✴ ✷✽

slide-53
SLIDE 53

❆♣♣❧✐❝❛t✐♦♥s

■♥❞❡①

✶

■♥tr♦❞✉❝t✐♦♥

✷

▼❛✐♥ ❘❡s✉❧ts

✸

❇❛s✐❝ ❘❡s✉❧ts

✹

❆♣♣❧✐❝❛t✐♦♥s

✷✷ ✴ ✷✽

slide-54
SLIDE 54

❆♣♣❧✐❝❛t✐♦♥s

❚♦♣♦❧♦❣✐❝❛❧ ●r♦✉♣s

❈♦r♦❧❧❛r② ✸ ▲❡t X ❛♥❞ (M, d) ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣ ❛♥❞ ❛ ♠❡tr✐❝ s❡♣❛r❛❜❧❡ ❣r♦✉♣✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ ❧❡t G ❜❡ ❛ s✉❜s❡t ♦❢ CHom(X, M) s✉❝❤ t❤❛t G

MX

✐s ❝♦♠♣❛❝t✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s✿ ✭❛✮ G ✐s ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❜✮ G ✐s r❡❧❛t✐✈❡❧② ❝♦♠♣❛❝t ✐♥ CHom(X, M) ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝♦♠♣❛❝t ♦♣❡♥ t♦♣♦❧♦❣②✳ ✭❝✮ ❚❤❡r❡ ❡①✐sts ❛ ❞❡♥s❡ ❇❛✐r❡ s✉❜s❡t F ⊆ X s✉❝❤ t❤❛t (G

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✳ ✭❞✮ ❚❤❡r❡ ❡①✐sts ❛ ❞❡♥s❡ Gδ s✉❜s❡t F ⊆ X s✉❝❤ t❤❛t (F, tp(G

MX

)) ✐s ▲✐♥❞❡❧ö❢✳ ❚❤❡♥ (d) ⇒ (c) ⇒ (a) ⇔ (b)✳ ■❢ X ✐s ❛❧s♦ ω✲♥❛rr♦✇✱ t❤❡♥ ❛❧❧ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ❋✉rt❤❡r♠♦r❡ (c) ❛♥❞ (d) ❛r❡ ❛❧s♦ tr✉❡ ❢♦r F = X✳

✷✷ ✴ ✷✽

slide-55
SLIDE 55

❆♣♣❧✐❝❛t✐♦♥s

❚♦♣♦❧♦❣✐❝❛❧ ●r♦✉♣s

❉❡✜♥✐t✐♦♥ ❆ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣ G ✐s s❛✐❞ t♦ ❜❡ ω✲♥❛rr♦✇ ✐❢ ❢♦r ❡✈❡r② ♥❡✐❣❤❜♦r❤♦♦❞ V ♦❢ t❤❡ ♥❡✉tr❛❧ ❡❧❡♠❡♥t✱ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t E ♦❢ G s✉❝❤ t❤❛t G = EV ✳

✷✷ ✴ ✷✽

slide-56
SLIDE 56

❆♣♣❧✐❝❛t✐♦♥s

❚♦♣♦❧♦❣✐❝❛❧ ●r♦✉♣s

❈♦r♦❧❧❛r② ✸ ▲❡t X ❛♥❞ (M, d) ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣ ❛♥❞ ❛ ♠❡tr✐❝ s❡♣❛r❛❜❧❡ ❣r♦✉♣✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ ❧❡t G ❜❡ ❛ s✉❜s❡t ♦❢ CHom(X, M) s✉❝❤ t❤❛t G

MX

✐s ❝♦♠♣❛❝t✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s✿ ✭❛✮ G ✐s ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❜✮ G ✐s r❡❧❛t✐✈❡❧② ❝♦♠♣❛❝t ✐♥ CHom(X, M) ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝♦♠♣❛❝t ♦♣❡♥ t♦♣♦❧♦❣②✳ ✭❝✮ ❚❤❡r❡ ❡①✐sts ❛ ❞❡♥s❡ ❇❛✐r❡ s✉❜s❡t F ⊆ X s✉❝❤ t❤❛t (G

MX

)|F ✐s ♠❡tr✐③❛❜❧❡✳ ✭❞✮ ❚❤❡r❡ ❡①✐sts ❛ ❞❡♥s❡ Gδ s✉❜s❡t F ⊆ X s✉❝❤ t❤❛t (F, tp(G

MX

)) ✐s ▲✐♥❞❡❧ö❢✳ ❚❤❡♥ (d) ⇒ (c) ⇒ (a) ⇔ (b)✳ ■❢ X ✐s ❛❧s♦ ω✲♥❛rr♦✇✱ t❤❡♥ ❛❧❧ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ❋✉rt❤❡r♠♦r❡ (c) ❛♥❞ (d) ❛r❡ ❛❧s♦ tr✉❡ ❢♦r F = X✳

✷✷ ✴ ✷✽

slide-57
SLIDE 57

❆♣♣❧✐❝❛t✐♦♥s

❚♦♣♦❧♦❣✐❝❛❧ ●r♦✉♣s

❈♦r♦❧❧❛r② ✹ ▲❡t X ❛♥❞ (M, d) ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣ ❛♥❞ ❛ ♠❡tr✐❝ ❣r♦✉♣✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ ❧❡t G ❜❡ ❛ s✉❜s❡t ♦❢ CHom(X, M) s✉❝❤ t❤❛t G

MX

✐s ❝♦♠♣❛❝t✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✭❛✮ G ✐s ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❜✮ L ✐s ❡q✉✐❝♦♥t✐♥✉♦✉s ♦♥ F✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ✭❝✮ ((L

MX

)|F, tp(F)) ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ✭❞✮ (F, tp(L

MX

)) ✐s ▲✐♥❞❡❧ö❢✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳

✷✸ ✴ ✷✽

slide-58
SLIDE 58

❆♣♣❧✐❝❛t✐♦♥s

❉②♥❛♠✐❝❛❧ ❙②st❡♠s

❈♦r♦❧❧❛r② ✺ ▲❡t X ❜❡ ❛ P♦❧✐s❤ G✲s♣❛❝❡ s✉❝❤ t❤❛t G

X X

✐s ❝♦♠♣❛❝t✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✭❛✮ X ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❜✮ ❚❤❡r❡ ❡①✐sts ❛ ❞❡♥s❡ ❇❛✐r❡ s✉❜s❡t F ⊆ X s✉❝❤ t❤❛t (G

X X

)|F ✐s ♠❡tr✐③❛❜❧❡✳ ✭❝✮ ❚❤❡r❡ ❡①✐sts ❛ ❞❡♥s❡ Gδ s✉❜s❡t F ⊆ X s✉❝❤ t❤❛t (F, tp(G

X X

)) ✐s ▲✐♥❞❡❧ö❢✳

✷✹ ✴ ✷✽

slide-59
SLIDE 59

❆♣♣❧✐❝❛t✐♦♥s

❉②♥❛♠✐❝❛❧ ❙②st❡♠s

❈♦r♦❧❧❛r② ✺ ▲❡t X ❜❡ ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐③❛❜❧❡ G✲s♣❛❝❡ s✉❝❤ t❤❛t G

X X

✐s ❝♦♠♣❛❝t✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✭❛✮ X ✐s ❤❡r❡❞✐t❛r② ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ ✭❜✮ L ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ♦♥ F✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ✭❝✮ ((L

MX

)|F, tp(F)) ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ✭❞✮ (F, tp(L

MX

)) ✐s ▲✐♥❞❡❧ö❢✱ ❢♦r ❛❧❧ L ∈ [G]≤ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳

✷✺ ✴ ✷✽

slide-60
SLIDE 60

❆♣♣❧✐❝❛t✐♦♥s

❆❦✐♥✱ ❊✳✱ ❆✉s❧❛♥❞❡r✱ ❏✳✱ ❛♥❞ ❇❡r❣✱ ❑✳ ✭✶✾✾✽✮✳ ❆❧♠♦st ❡q✉✐❝♦♥t✐♥✉✐t② ❛♥❞ t❤❡ ❡♥✈❡❧♦♣✐♥❣ s❡♠✐❣r♦✉♣✳ ❈♦♥t❡♠♣✳ ▼❛t❤✳✱ ✷✶✺✿✼✺✕✽✶✳ ❈❛s❝❛❧❡s✱ ❇✳✱ ◆❛♠✐♦❦❛✱ ■✳✱ ❛♥❞ ❱❡r❛✱ ●✳ ✭✷✵✵✵✮✳ ❚❤❡ ▲✐♥❞❡❧ö❢ ♣r♦♣❡rt② ❛♥❞ ❢r❛❣♠❡♥t❛❜✐❧✐t②✳ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✷✽✭✶✶✮✿✸✸✵✶✕✸✸✵✾✳ ❈❤r✐st❡♥s❡♥✱ ❏✳ ✭✶✾✽✶✮✳ ❏♦✐♥t ❈♦♥t✐♥✉✐t② ♦❢ s❡♣❛r❛t❡❧② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✽✷✭✸✮✿✹✺✺✕✹✻✶✳ ❊♥❣❡❧❦✐♥❣✱ ❘✳ ✭✶✾✽✾✮✳

  • ❡♥❡r❛❧ t♦♣♦❧♦❣②✳

❍❡❧❞❡r♠❛♥♥ ❱❡r❧❛❣✳

  • ❧❛s♥❡r✱ ❊✳ ❛♥❞ ▼❡❣r❡❧✐s❤✈✐❧✐✱ ▼✳ ✭✷✵✵✻✮✳

❍❡r❡❞✐t❛r✐❧② ♥♦♥✲s❡♥s✐t✐✈❡ ❞②♥❛♠✐❝❛❧ s②st❡♠s ❛♥❞ ❧✐♥❡❛r r❡♣r❡s❡♥t❛t✐♦♥s✳ ❈♦❧❧♦q✳ ▼❛t❤✳✱ ✶✵✹✭✷✮✿✷✷✸✕✷✽✸✳

✷✻ ✴ ✷✽

slide-61
SLIDE 61

❆♣♣❧✐❝❛t✐♦♥s

  • ❧❛s♥❡r✱ ❊✳ ▲✳ ■✳✱ ▼❡❣r❡❧✐s❤✈✐❧✐✱ ▼✳✱ ❛♥❞ ❯s♣❡♥s❦✐❥✱ ❱✳ ❱✳ ✭✷✵✵✻✮✳

❖♥ ♠❡tr✐③❛❜❧❡ ❡♥✈❡❧♦♣✐♥❣ s❡♠✐❣r♦✉♣s✳ ■sr❛❡❧ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✶✻✹✭✶✮✿✸✶✼✕✸✸✷✳ ❚r♦❛❧❧✐❝✱ ❏✳ ✭✶✾✾✻✮✳ ❙❡q✉❡♥t✐❛❧ ❝r✐t❡r✐❛ ❢♦r ❡q✉✐❝♦♥t✐♥✉✐t② ❛♥❞ ✉♥✐❢♦r♠✐t✐❡s ♦♥ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣s✳ ❚♦♣♦❧♦❣② ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✻✽✭✶✮✿✽✸✕✾✺✳

✷✼ ✴ ✷✽

slide-62
SLIDE 62

❆♣♣❧✐❝❛t✐♦♥s

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✷✽ ✴ ✷✽

slide-63
SLIDE 63

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

■♥❞❡①

✺

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

✻

Pr♦♦❢ ♦❢ ⋆

✷✾ ✴ ✷✽

slide-64
SLIDE 64

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢ ✿ ❈♦♥s✐❞❡r t❤❡ ✭❝♦♥t✐♥✉♦✉s✮ ♠❛♣ νF : (G

MX

)|F × K → [−✶, ✶]F ❞❡✜♥❡❞ ❜② νF(h, α) ❞❡❢ = α ◦ h ❢♦r ❛❧❧ h ∈ (G

MX

)|F ❛♥❞ α ∈ K✳

✷✽ ✴ ✷✽

slide-65
SLIDE 65

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢ ✿ ❈♦♥s✐❞❡r t❤❡ ✭❝♦♥t✐♥✉♦✉s✮ ♠❛♣ νF : (G

MX

)|F × K → [−✶, ✶]F ❞❡✜♥❡❞ ❜② νF(h, α) ❞❡❢ = α ◦ h ❢♦r ❛❧❧ h ∈ (G

MX

)|F ❛♥❞ α ∈ K✳ νF(G|F × K) ✐s ❛ s✉❜s❡t ♦❢ C(F, [−✶, ✶]) s✉❝❤ t❤❛t✿ ✭✐✮ νF(G|F × K) = ν(G × K)|F ✭✐✐✮ (ν(G × K)

[−,✶,✶]X

)|F = νF((G

MX

)|F × K) = ν(G

MX

× K)|F

✷✽ ✴ ✷✽

slide-66
SLIDE 66

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢ ✿ ❈♦♥s✐❞❡r t❤❡ ✭❝♦♥t✐♥✉♦✉s✮ ♠❛♣ νF : (G

MX

)|F × K → [−✶, ✶]F ❞❡✜♥❡❞ ❜② νF(h, α) ❞❡❢ = α ◦ h ❢♦r ❛❧❧ h ∈ (G

MX

)|F ❛♥❞ α ∈ K✳ νF(G|F × K) ✐s ❛ s✉❜s❡t ♦❢ C(F, [−✶, ✶]) s✉❝❤ t❤❛t✿ ✭✐✮ νF(G|F × K) = ν(G × K)|F ✭✐✐✮ (ν(G × K)

[−,✶,✶]X

)|F = νF((G

MX

)|F × K) = ν(G

MX

× K)|F (F, tp(νF((G

MX

)|F × K))) ✐s ▲✐♥❞❡❧ö❢✳

✷✽ ✴ ✷✽

slide-67
SLIDE 67

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢ ✿ ❈♦♥s✐❞❡r t❤❡ ✭❝♦♥t✐♥✉♦✉s✮ ♠❛♣ νF : (G

MX

)|F × K → [−✶, ✶]F ❞❡✜♥❡❞ ❜② νF(h, α) ❞❡❢ = α ◦ h ❢♦r ❛❧❧ h ∈ (G

MX

)|F ❛♥❞ α ∈ K✳ νF(G|F × K) ✐s ❛ s✉❜s❡t ♦❢ C(F, [−✶, ✶]) s✉❝❤ t❤❛t✿ ✭✐✮ νF(G|F × K) = ν(G × K)|F ✭✐✐✮ (ν(G × K)

[−,✶,✶]X

)|F = νF((G

MX

)|F × K) = ν(G

MX

× K)|F (F, tp(νF((G

MX

)|F × K))) ✐s ▲✐♥❞❡❧ö❢✳ ❘❡❛s♦♥✐♥❣ ❜② ❝♦♥tr❛❞✐❝t✐♦♥✱ s✉♣♣♦s❡ t❤❛t ν(G × K) ✐s ♥♦t ❆❊✳

✷✽ ✴ ✷✽

slide-68
SLIDE 68

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢ ✿ ❈♦♥s✐❞❡r t❤❡ ✭❝♦♥t✐♥✉♦✉s✮ ♠❛♣ νF : (G

MX

)|F × K → [−✶, ✶]F ❞❡✜♥❡❞ ❜② νF(h, α) ❞❡❢ = α ◦ h ❢♦r ❛❧❧ h ∈ (G

MX

)|F ❛♥❞ α ∈ K✳ νF(G|F × K) ✐s ❛ s✉❜s❡t ♦❢ C(F, [−✶, ✶]) s✉❝❤ t❤❛t✿ ✭✐✮ νF(G|F × K) = ν(G × K)|F ✭✐✐✮ (ν(G × K)

[−,✶,✶]X

)|F = νF((G

MX

)|F × K) = ν(G

MX

× K)|F (F, tp(νF((G

MX

)|F × K))) ✐s ▲✐♥❞❡❧ö❢✳ ❘❡❛s♦♥✐♥❣ ❜② ❝♦♥tr❛❞✐❝t✐♦♥✱ s✉♣♣♦s❡ t❤❛t ν(G × K) ✐s ♥♦t ❆❊✳ ❇② ▲❡♠♠❛ ✶ t❤❡r❡ ❡①✐sts ❛ ❝♦♠♣❛❝t ❛♥❞ s❡♣❛r❛❜❧❡ s✉❜s❡t KF ♦❢ F✱ ❛♥ ♦♥t♦ ❛♥❞ ❝♦♥t✐♥✉♦✉s ♠❛♣ Ψ : KF → ✷ω✱ ❛♥❞ ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t L ♦❢ ν(G × K) s✉❝❤ t❤❛t t❤❡ s✉❜s❡t L∗ ⊆ C(✷ω, [−✶, ✶]) ❞❡✜♥❡❞ ❜② l∗(Ψ(x)) = l(x) ❢♦r ❛❧❧ x ∈ KF ✐s ♥♦t ❆❊✳

✷✽ ✴ ✷✽

slide-69
SLIDE 69

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢ ✿ (KF, tp(νKF ((G

MX

)|KF × K))) ✐s ▲✐♥❞❡❧ö❢✳

✷✽ ✴ ✷✽

slide-70
SLIDE 70

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢ ✿ (KF, tp(νKF ((G

MX

)|KF × K))) ✐s ▲✐♥❞❡❧ö❢✳ (✷ω, tp(L∗[−✶,✶]✷ω )) ✐s ❛❧s♦ ▲✐♥❞❡❧ö❢✳

✷✽ ✴ ✷✽

slide-71
SLIDE 71

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢ ✿ (KF, tp(νKF ((G

MX

)|KF × K))) ✐s ▲✐♥❞❡❧ö❢✳ (✷ω, tp(L∗[−✶,✶]✷ω )) ✐s ❛❧s♦ ▲✐♥❞❡❧ö❢✳ ❇② ❈◆❱✱ L∗ ✐s ❛♥ ❍❆❊ ❢❛♠✐❧② ♦♥ ✷ω✳ ❈♦♥tr❛❞✐❝t✐♦♥

✷✽ ✴ ✷✽

slide-72
SLIDE 72

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢ ✿ (KF, tp(νKF ((G

MX

)|KF × K))) ✐s ▲✐♥❞❡❧ö❢✳ (✷ω, tp(L∗[−✶,✶]✷ω )) ✐s ❛❧s♦ ▲✐♥❞❡❧ö❢✳ ❇② ❈◆❱✱ L∗ ✐s ❛♥ ❍❆❊ ❢❛♠✐❧② ♦♥ ✷ω✳ ❈♦♥tr❛❞✐❝t✐♦♥ ❇② ❈♦r♦❧❧❛r② ✶✱ G ✐s ❆❊✳

✷✽ ✴ ✷✽

slide-73
SLIDE 73

Pr♦♦❢ ♦❢ ⋆

■♥❞❡①

✺

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

✻

Pr♦♦❢ ♦❢ ⋆

✷✾ ✴ ✷✽

slide-74
SLIDE 74

Pr♦♦❢ ♦❢ ⋆

▲❡t L ❜❡ ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t ♦❢ G ⊆ C(X, M)✳ ❲❡ ❞❡♥♦t❡ ❜② XL t❤❡ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ( X, tp(˜ L))✱ ✇❤✐❝❤ ✐s ♠❡tr✐③❛❜❧❡ ❜❡❝❛✉s❡ ˜ L ✐s ❝♦✉♥t❛❜❧❡✳ ❈♦♥s✐❞❡r t❤❡ ♠❛♣ p∗ : (M ˜

X, tp( ˜

X)) → (MX, tp(X)) ❞❡✜♥❡❞ ❜② p∗(˜ f ) = ˜ f ◦ p✱ ❢♦r ❡❛❝❤ ˜ f ∈ M ˜

X✳

Pr♦♣♦s✐t✐♦♥ p∗ ✐s ❛ ❤♦♠❡♦♠♦r♣❤✐s♠ ♦❢ ˜ L

M ˜

X

♦♥t♦ L

MX

✳ Pr♦♦❢ ✿ ❲❡ ♦❜s❡r✈❡ t❤❛t p∗ ✐s ❝♦♥t✐♥✉♦✉s✱ s✐♥❝❡ ❛ ♥❡t { ˜ fα}α∈A tp( ˜ X)✲❝♦♥✈❡r❣❡s t♦ ˜ f ✐♥ ˜ L

M ˜

X

✐❢ ❛♥❞ ♦♥❧② ✐❢ { ˜ fα ◦ p}α∈A tp(X)✲❝♦♥✈❡r❣❡s t♦ ˜ f ◦ p ✐♥ L

MX

✳ ▲❡t✬s s❡❡ t❤❛t p∗(˜ L

M ˜

X

) = L

MX

✳ ■♥❞❡❡❞✱ s✐♥❝❡ p∗ ✐s ❝♦♥t✐♥✉♦✉s ✇❡ ❤❛✈❡ t❤❛t p∗(˜ L

M ˜

X

) ⊆ p∗(˜ L)

MX

= L

MX

✳ ❲❡ ❤❛✈❡ t❤❡ ♦t❤❡r ✐♥❝❧✉s✐♦♥ ❜❡❝❛✉s❡ L

MX

✐s t❤❡ s♠❛❧❧❡r ❝❧♦s❡❞ s❡t t❤❛t ❝♦♥t❛✐♥s L ❛♥❞ L ⊆ p∗(˜ L

M ˜

X

)✳

✷✾ ✴ ✷✽

slide-75
SLIDE 75

Pr♦♦❢ ♦❢ ⋆

Pr♦♦❢ ✿ ▲❡t ˜ f , ˜ g ∈ ˜ L

M ˜

X

s✉❝❤ t❤❛t ˜ f = ˜ g✱ t❤❡♥ t❤❡r❡ ❡①✐st ˜ x ∈ ˜ X s✉❝❤ t❤❛t ˜ f (˜ x) = ˜ g(˜ x)✳ ▲❡t x ∈ X ❛♥ ❡❧❡♠❡♥t s✉❝❤ t❤❛t ˜ x = p(x)✱ t❤❡♥ (˜ f ◦ p)(x) = (˜ g ◦ p)(x)✳ ❙♦✱ p∗ ✐s ✐♥❥❡❝t✐✈❡ ❜❡❝❛✉s❡ ˜ f ◦ p = ˜ g ◦ p✳ ❋✐♥❛❧❧②✱ ✇❡ ❛rr✐✈❡ t♦ t❤❡ ❝♦♥❝❧✉s✐♦♥ t❤❛t p∗|

˜ L

M ˜ X ✐s ❛ ❤♦♠❡♦♠♦r♣❤✐s♠

❜❡❝❛✉s❡ ✐t ✐s ❞❡✜♥❡❞ ❜❡t✇❡❡♥ ❝♦♠♣❛❝t s♣❛❝❡s✳

✸✵ ✴ ✷✽