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❊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

■♥❞❡① ■♥tr♦❞✉❝t✐♦♥ ✶ ▼❛✐♥ ❘❡s✉❧ts ✷ ❇❛s✐❝ ❘❡s✉❧ts ✸ ❆♣♣❧✐❝❛t✐♦♥s ✹

■♥tr♦❞✉❝t✐♦♥ ■♥❞❡① ■♥tr♦❞✉❝t✐♦♥ ✶ ▼❛✐♥ ❘❡s✉❧ts ✷ ❇❛s✐❝ ❘❡s✉❧ts ✸ ❆♣♣❧✐❝❛t✐♦♥s ✹ ✶ ✴ ✷✽

■♥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 ✳ ✶ ✴ ✷✽

■♥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 ✭❜✮✳ ✷ ✴ ✷✽

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

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

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

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

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

■♥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 ❍❆❊✳ ✸ ✴ ✷✽

▼❛✐♥ ❘❡s✉❧ts ■♥❞❡① ■♥tr♦❞✉❝t✐♦♥ ✶ ▼❛✐♥ ❘❡s✉❧ts ✷ ❇❛s✐❝ ❘❡s✉❧ts ✸ ❆♣♣❧✐❝❛t✐♦♥s ✹ ✹ ✴ ✷✽

▼❛✐♥ ❘❡s✉❧ts ❚❤❡♦r❡♠ ❆ ▲❡t X ❛♥❞ ( M , d ) ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ s♣❛❝❡ ❛♥❞ ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ M X s♣❛❝❡✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ ❧❡t G ⊆ C ( X , M ) s✉❝❤ t❤❛t G ✐s ❝♦♠♣❛❝t✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s✿ ✭❛✮ G ✐s ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s✳ M X ✭❜✮ ❚❤❡r❡ ❡①✐sts ❛ ❞❡♥s❡ ❇❛✐r❡ s✉❜s❡t F ⊆ X s✉❝❤ t❤❛t ( G ) | F ✐s ♠❡tr✐③❛❜❧❡✳ M X ✭❝✮ ❚❤❡r❡ ❡①✐sts ❛ ❞❡♥s❡ G δ s✉❜s❡t F ⊆ X s✉❝❤ t❤❛t ( F , t p ( G )) ✐s ▲✐♥❞❡❧ö❢✳ ❚❤❡♥ ( b ) ⇒ ( c ) ⇒ ( a ) ✳ ■❢ X ✐s ❛❧s♦ ❛ ❤❡r❡❞✐t❛r✐❧② ▲✐♥❞❡❧ö❢ s♣❛❝❡✱ t❤❡♥ ❛❧❧ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ✹ ✴ ✷✽

▼❛✐♥ ❘❡s✉❧ts ❚❤❡♦r❡♠ ❇ ▲❡t X ❛♥❞ ( M , d ) ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ s♣❛❝❡ ❛♥❞ ❛ ♠❡tr✐❝ s♣❛❝❡✱ M X r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ ❧❡t G ⊆ C ( X , M ) s✉❝❤ t❤❛t G ✐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 ✳ M X ) | F ✐s ♠❡tr✐③❛❜❧❡✱ ❢♦r ❛❧❧ L ∈ [ G ] ≤ ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ✭❝✮ ( L ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X ✳ M X )) ✐s ▲✐♥❞❡❧ö❢✱ ❢♦r ❛❧❧ L ∈ [ G ] ≤ ω ❛♥❞ F ❛ s❡♣❛r❛❜❧❡ ❛♥❞ ✭❞✮ ( F , t p ( L ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X ✳ ✺ ✴ ✷✽

❇❛s✐❝ ❘❡s✉❧ts ■♥❞❡① ■♥tr♦❞✉❝t✐♦♥ ✶ ▼❛✐♥ ❘❡s✉❧ts ✷ ❇❛s✐❝ ❘❡s✉❧ts ✸ ❆♣♣❧✐❝❛t✐♦♥s ✹ ✻ ✴ ✷✽

❇❛s✐❝ ❘❡s✉❧ts ❉❡✜♥✐t✐♦♥ X ✐s s❛✐❞ t♦ ❜❡ ❤❡♠✐❝♦♠♣❛❝t ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❛❝ts s❡ts { X n } n ∈ ω s✉❝❤ t❤❛t X = � X n ❛♥❞ ❢♦r ❡✈❡r② ❝♦♠♣❛❝t s✉❜s❡t C ♦❢ X t❤❡r❡ n ∈ ω ✐s n ∈ ω s✉❝❤ t❤❛t C ⊆ X n ✳ ✻ ✴ ✷✽

� � � ❇❛s✐❝ ❘❡s✉❧ts ▲❡♠♠❛ ✶ ▲❡t X ❛♥❞ ( M , d ) ❜❡ ❛ ❷❡❝❤✲❝♦♠♣❧❡t❡ s♣❛❝❡ ❛♥❞ ❛ ❤❡♠✐❝♦♠♣❛❝t ♠❡tr✐❝ M X s♣❛❝❡✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ ❧❡t G ❜❡ ❛ s✉❜s❡t ♦❢ C ( X , M ) s✉❝❤ t❤❛t G ✐s ❝♦♠♣❛❝t✳ ■❢ ● ✐s ♥♦t ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ✱ t❤❡♥ ❢♦r ❡✈❡r② G δ ❛♥❞ ❞❡♥s❡ s✉❜s❡t F ♦❢ X t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ s✉❜s❡t L ✐♥ G ✱ ❛ ❝♦♠♣❛❝t s❡♣❛r❛❜❧❡ s✉❜s❡t K F ⊆ F ✱ ❛ ❝♦♠♣❛❝t s✉❜s❡t N ⊆ M ❛♥❞ ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ s✉r❥❡❝t✐✈❡ ♠❛♣ Ψ ♦❢ K F ♦♥t♦ t❤❡ ❈❛♥t♦r s❡t ✷ ω s✉❝❤ t❤❛t ✐❢ t❤❡ ♠❛♣s l ∗ ❛r❡ ❞❡✜♥❡❞ t♦ ♠❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠ ❝♦♠♠✉t❛t✐✈❡ Ψ ✷ ω K F l | KF l ∗ N t❤❡♥ t❤❡ s✉❜s❡t ▲ ∗ := { ❧ ∗ : ❧ ∈ ▲ } ⊆ C ( ✷ ω , N ) ✐s ♥♦t ❛❧♠♦st ❡q✉✐❝♦♥t✐♥✉♦✉s ♦♥ ✷ ω ✳ ✼ ✴ ✷✽