Solutions to the Exercises in Lecture Notes on Decision Theory Gil Riella October 14, 2016

❙♦❧✉t✐♦♥s t♦ t❤❡ ❊①❡r❝✐s❡s ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ♦♥

❉❡❝✐s✐♦♥ ❚❤❡♦r②

❈♦♥t❡♥ts

❈❤❛♣t❡r ✶✳ ❇✐♥❛r② ❘❡❧❛t✐♦♥s ✹

❈❤❛♣t❡r ✷✳ ❘❡✈❡❛❧❡❞ Pr❡❢❡r❡♥❝❡ ❚❤❡♦r② ✼

❈❤❛♣t❡r ✸✳ ❘❡♣r❡s❡♥t❛t✐♦♥ ❜② ❯t✐❧✐t② ❋✉♥❝t✐♦♥ ✶✷

❈❤❛♣t❡r ✹✳ ❊①♣❡❝t❡❞ ❯t✐❧✐t② ❚❤❡♦r② ✶✺

❈❤❛♣t❡r ✺✳ ◆♦♥✲❊①♣❡❝t❡❞ ❯t✐❧✐t② ❚❤❡♦r② ✷✶

❈❤❛♣t❡r ✻✳ ❈♦♠♣❧❡t❡ Pr❡❢❡r❡♥❝❡s ❯♥❞❡r ❯♥❝❡rt❛✐♥t② ✷✷

❈❤❛♣t❡r ✼✳ ■♥❝♦♠♣❧❡t❡ Pr❡❢❡r❡♥❝❡s ✉♥❞❡r ❯♥❝❡rt❛✐♥t② ✷✹

❈❤❛♣t❡r ✽✳ ❯♥❝❡rt❛✐♥t② ❆✈❡rs❡ Pr❡❢❡r❡♥❝❡s ✷✺

❈❤❛♣t❡r ✾✳ Pr❡❢❡r❡♥❝❡s ♦✈❡r ▼❡♥✉s ✷✻

❆♣♣❡♥❞✐① ❆✳ ❯s❡❢✉❧ ▼❛t❤❡♠❛t✐❝❛❧ ❘❡s✉❧ts ✷✼

❇✐❜❧✐♦❣r❛♣❤② ✷✽

❈❍❆P❚❊❘ ✶

❇✐♥❛r② ❘❡❧❛t✐♦♥s

❊①❡r❝✐s❡ ✶✳✶✳ ❙❤♦✇ t❤❛t✱ ❢♦r ❡✈❡r② ❜✐♥❛r② r❡❧❛t✐♦♥ %✱ _≻ ✐s ❛s②♠♠❡tr✐❝✱ ✐✳❡✳ x _≻y ✐♠♣❧✐❡s ✐t

✐s ♥♦t tr✉❡ t❤❛t y≻x✱ ❛♥❞ ∼ ✐s s②♠♠❡tr✐❝✳

❙♦❧✉t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t x ≻ y✳ ❚❤✐s ♠❡❛♥s t❤❛t x % y✱ ❜✉t ✐t ✐s ♥♦t tr✉❡ t❤❛t y % x✳ ❇✉t

t❤❡♥ ✐t ✐s ❝❧❡❛r t❤❛t ✐t ❝❛♥♥♦t ❜❡ t❤❡ ❝❛s❡ t❤❛t y % x✱ ❜✉t ✐t ✐s ♥♦t tr✉❡ t❤❛t x % y✱ ✇❤✐❝❤ ✐s t❤❡

❞❡✜♥✐t✐♦♥ ♦❢y≻x✳ ❋✐♥❛❧❧②✱ s✉♣♣♦s❡ t❤❛tx∼y✳ ❇② ❞❡✜♥✐t✐♦♥✱ t❤✐s ♠❡❛♥s t❤❛tx%y❛♥❞ ✐t ✐s ♥♦t

t❤❡ ❝❛s❡ t❤❛t x _≻y✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t y% x✳ ❙✐♥❝❡ x %y✱ ✇❡ ♦❜✈✐♦✉s❧② ❞♦ ♥♦t ❤❛✈❡ t❤❛t y _≻x✱

s♦ t❤❛t y_∼x✳

❊①❡r❝✐s❡ ✶✳✷✳ ❙❤♦✇ t❤❛t ✐❢ % ✐s ❛ ♣r❡♦r❞❡r✱ t❤❡♥ _≻ ✐s tr❛♥s✐t✐✈❡ ❛♥❞ _∼ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡

r❡❧❛t✐♦♥✳

❙♦❧✉t✐♦♥✳ ❙✉♣♣♦s❡ x≻ y ❛♥❞ y ≻z✳ ❚❤✐s ♠❡❛♥s t❤❛t x% y✱ ❜✉t ✐t ✐s ♥♦t tr✉❡ t❤❛t y% x✱

❛♥❞ y%z✱ ❜✉t ✐t ✐s ♥♦t tr✉❡ t❤❛t z %y✳ ❚r❛♥s✐t✐✈✐t② ♦❢% ✐♠♣❧✐❡s t❤❛tx%z✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱

✐❢ z %x✱ t❤❡♥ tr❛♥s✐t✐✈✐t② ♦❢ %✇♦✉❧❞ ✐♠♣❧② t❤❛t z %y✱ ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts ♦✉r ✐♥✐t✐❛❧ ❛ss✉♠♣t✐♦♥✳

❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ✐t ✐s ♥♦t tr✉❡ t❤❛t z %x ❛♥❞✱ t❤❡r❡❢♦r❡✱x≻z✳

◆♦✇ ♣✐❝❦ ❛♥② x ∈ X✳ ❙✐♥❝❡ % ✐s r❡✢❡①✐✈❡✱ ✇❡ ❤❛✈❡ x %x✳ ❙✐♥❝❡✱ ♦❜✈✐♦✉s❧②✱ ✐t ❝❛♥♥♦t ❜❡ t❤❡

❝❛s❡ t❤❛t x _≻ x✱ t❤✐s ✐♠♣❧✐❡s t❤❛t x _∼ x✳ ◆♦✇ s✉♣♣♦s❡ t❤❛t x _∼ y✳ ❚❤✐s ♠❡❛♥s t❤❛t x % y ❛♥❞

✐t ✐s ♥♦t tr✉❡ t❤❛t x _≻ y✳ ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ s❛② t❤❛t x % y ❛♥❞ y % x✳ ■t ✐s ❝❧❡❛r ♥♦✇ t❤❛t x_∼y✐♠♣❧✐❡s y_∼x✳✶ ❋✐♥❛❧❧②✱ s✉♣♣♦s❡ t❤❛tx_∼y ❛♥❞y _∼z✳ ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ s❛② t❤❛tx%y✱ y %x✱ y % z ❛♥❞ z %y✳ ❇✉t tr❛♥s✐t✐✈✐t② ♦❢ % ✐♠♣❧✐❡s t❤❛t x %z ❛♥❞ z % x✱ ✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t

t♦ s❛② t❤❛t x_∼z✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t _∼ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳

❊①❡r❝✐s❡ ✶✳✸✳ ❙❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣✿

✭❛✮ ❚❤❡ ❜✐♥❛r② r❡❧❛t✐♦♥ tran(%) ✐s tr❛♥s✐t✐✈❡✳

✭❜✮ ■❢_%ˆ _{✐s ❛ tr❛♥s✐t✐✈❡ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦♥X ❛♥❞ %⊆%}ˆ_{✱ t❤❡♥ tran(%)⊆%}ˆ_{✳ ✭❚❤❛t ✐s✱tran(%)}

✐s t❤❡ s♠❛❧❧❡st tr❛♥s✐t✐✈❡ r❡❧❛t✐♦♥ s✉❝❤ t❤❛t%_⊆tran(%)✮✳

❙♦❧✉t✐♦♥✳

✭❛✮ ❙✉♣♣♦s❡ t❤❛txtran(%)y❛♥❞ytran(%)z✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡r❡ ❡①✐st s❡q✉❡♥❝❡sz1, ..., zm

❛♥❞ zˆ1, ...,znˆ s✉❝❤ t❤❛t x % z1✱ z1 % z2✱ ✳✳✳✱ zm−1 % zm✱ zm % y ❛♥❞ y % zˆ1✱ zˆ1 % zˆ2✱ ✳✳✳✱

ˆ

zn−1 % znˆ ✱ znˆ % z✳ ❇✉t t❤❡♥✱ ✐❢ ✇❡ ❞❡✜♥❡ wi := zi✱ ❢♦r i = 1, ..., m✱ wm+1 := y ❛♥❞

wi := ˆzi−(m+1) ❢♦r i=m+ 2, ..., m+n+ 1✱ ✇❡ ❣❡t x %w1✱ w1 % w2✱ ✳✳✳✱ wm+n% wm+n+1

❛♥❞ wm+n+1 %z✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t xtran(%)z✳

✭❜✮ ❙✉♣♣♦s❡ t❤❛t _%ˆ _{✐s tr❛♥s✐t✐✈❡ ❛♥❞ t❤❛t %⊆%}ˆ_{✳ ❋♦r ❛♥② x, y ∈X✱ ✐❢(x, y)∈tran(%)✱ t❤❡♥}

t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡z1, ..., zm ∈X s✉❝❤ t❤❛tx%z1✱z1 %z2✱ ✳✳✳✱zm−1 %zm ❛♥❞zm %y✳

❙✐♥❝❡ %_⊆%ˆ✱ t❤✐s ✐♠♣❧✐❡s t❤❛t x%ˆz1✱ z1%ˆz2✱ ✳✳✳✱ zm−1%ˆzm ❛♥❞ zm%ˆy✳ ❇② t❤❡ tr❛♥s✐t✐✈✐t②

♦❢ _%ˆ_{✱ t❤✐s ✐♠♣❧✐❡s t❤❛t x%}ˆ_{y✳ ❙✐♥❝❡ x ❛♥❞ y ✇❡r❡ ❡♥t✐r❡❧② ❣❡♥❡r✐❝ ❤❡r❡✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t}

tran(%)_⊆%ˆ✳

❊①❡r❝✐s❡ ✶✳✹✳ ❙✉♣♣♦s❡ t❤❛t (X,%) ✐s ❛ ♣r❡♦r❞❡r❡❞ s❡t ✭% ✐s ♥♦t ♥❡❝❡ss❛r✐❧② ❝♦♠♣❧❡t❡✮ ❛♥❞

|X|<∞✳ ❙❤♦✇ t❤❛t ▼❆❳(X,%)6=∅✳ ❍✐♥t✿ ❲❤❛t ❛❜♦✉t ❛♥ ✐♥❞✉❝t✐♦♥ ❛r❣✉♠❡♥t❄ ✶_{❚❤✐s ♥❡✇ ❡q✉✐✈❛❧❡♥t ❞❡✜♥✐t✐♦♥ ✐s ❡♥t✐r❡❧② s②♠♠❡tr✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛❧t❡r♥❛t✐✈❡sx❛♥❞y.}

✶✳ ❇■◆❆❘❨ ❘❊▲❆❚■❖◆❙ ✺ ❙♦❧✉t✐♦♥✳ ❋♦❧❧♦✇✐♥❣ t❤❡ ❤✐♥t✱ ❧❡t✬s tr② ❛♥ ✐♥❞✉❝t✐♦♥ ❛r❣✉♠❡♥t ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ♦❢

X✳ ❖❜✈✐♦✉s❧②✱ ▼❆❳(X,%)₆=_∅✐❢_|X_|= 1✳ ◆♦✇ s✉♣♣♦s❡ t❤❛t ▼❆❳(X,%)₆=_∅✇❤❡♥❡✈❡r_|X_|=n✱

❢♦r s♦♠❡ n _∈ N✳ ❋✐① s♦♠❡ s❡t X s✉❝❤ t❤❛t _|X_| =n+ 1 ❛♥❞ s✉♣♣♦s❡ t❤❛t % ✐s ❛ ♣r❡♦r❞❡r ♦♥ X✳

P✐❝❦ ❛♥② ❛❧t❡r♥❛t✐✈❡y_∈X ❛♥❞ ❝♦♥s✐❞❡r t❤❡ s❡tY :=X_{\ {}y}✳ ■t ✐s ♥♦t ❤❛r❞ t♦ s❡❡ t❤❛t(Y,%)✐s ❛

♣r❡♦r❞❡r❡❞ s❡t✳✷ _{❇② t❤❡ ✐♥❞✉❝t✐♦♥ ❤②♣♦t❤❡s✐s✱ t❤❡r❡ ❡①✐sts x}∗ _{∈Y s✉❝❤ t❤❛t ❢♦r ♥♦x∈Y✱ x≻x}∗_✳

■❢ ✐t ✐s ♥♦t tr✉❡ t❤❛t y _≻x∗_{✱ t❤❡♥✱ t❤❡r❡ ❞♦❡s ♥♦t ❡①✐st x∈X s✉❝❤ t❤❛t x ≻x}∗_{✱ ❛♥❞ ✇❡ ❝♦♥❝❧✉❞❡}

t❤❛t x∗ _{∈ ▼❆❳(X,%)✳ ■❢ y ≻ x}∗_{✱ t❤❡♥ tr❛♥s✐t✐✈✐t② ♦❢ ≻ ✐♠♣❧✐❡s t❤❛t✱ ❢♦r ♥♦ x ∈ Y✱ ✇❡ ❝❛♥}

❤❛✈❡ x _≻ y✳✸ ❚❤✐s ❝❧❡❛r❧② ✐♠♣❧✐❡s t❤❛t y _∈ ▼❆❳(X,%)✱ ❛♥❞ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ▼❆❳(X,%) ₆= _∅

✇❤❡♥❡✈❡r |X_|<_∞.

❊①❡r❝✐s❡ ✶✳✺ ✭❇❛s❡❞ ♦♥ ❖❦ ❛♥❞ ❘✐❡❧❧❛ ✭✷✵✶✹✮✮✳ ❙✉♣♣♦s❡%_⊆Rn_×Rn✐s ❛ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t

♣r❡♦r❞❡r✳ ❚❤❛t ✐s✱ s✉♣♣♦s❡ t❤❛t%✐s ❛ ♣r❡♦r❞❡r s✉❝❤ t❤❛tx%y✐♠♣❧✐❡s t❤❛tx+z %y+z ❢♦r ❡✈❡r② x, y, z _∈Rn_{✳ ❚❤❡ st❡♣s ❜❡❧♦✇ s❤♦✇ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❝♦♠♣❧❡t❡ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ♣r❡♦r❞❡r %}ˆ

t❤❛t ❡①t❡♥❞s %✳

✭❛✮ ❆❞❛♣t t❤❡ st❡♣s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❙③♣✐❧r❛❥♥✬s ❚❤❡♦r❡♠ t♦ s❤♦✇ t❤❛t t❤❡r❡ ❡①✐sts ❛ ♠❛①✐♠❛❧ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ♣r❡♦r❞❡r_%ˆ _{t❤❛t ❡①t❡♥❞s %✳}

✭❜✮ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ❡❧❡♠❡♥t x_∈Rn _{s✉❝❤ t❤❛tx ✐s ♥♦t%}ˆ_{✲❝♦♠♣❛r❛❜❧❡ t♦ 0✳ ❚❤❛t}

✐s✱ ♥❡✐t❤❡r x%ˆ0 ♥♦r 0 ˆ%x ❛r❡ tr✉❡✳ ❲❡ ✇✐❧❧ s❤♦✇ t❤❛t t❤✐s ✐♠♣❧✐❡s t❤❛t kx_≻ˆ0 ❢♦r s♦♠❡ k_∈N✳ ❋♦r t❤❛t✱ s✉♣♣♦s❡ t❤❛t ❢♦r ♥♦ k_∈N ✇❡ ❤❛✈❡kx_≻ˆ0✳ ❉❡✜♥❡0↑,%ˆ _:={y∈Rn_:y%ˆ_0}

❛♥❞ [[x]] := _{kx : k _∈ Z+}✳ ❋✐♥❛❧❧②✱ ❞❡✜♥❡ %∗ ❜② y %∗ z ⇐⇒ y−z ∈ (0↑,%ˆ −[[x]])✳✹

❙❤♦✇ t❤❛t %∗ _{✐s ❛ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ♣r❡♦r❞❡r t❤❛t ❡①t❡♥❞s %}ˆ_{✱ ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts ✐ts}

♠❛①✐♠❛❧✐t②✳ ❈♦♥❝❧✉❞❡ t❤❛tkx_≻ˆ0 ❢♦r s♦♠❡k _∈N✳

✭❝✮ ❯s❡ ❛♥ ❛r❣✉♠❡♥t ✈❡r② s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ✐♥ ♣❛rt ✭❜✮ t♦ s❤♦✇ t❤❛t 0 ˆ_≻kx ❢♦r s♦♠❡ k_∈N✳

✭❞✮ ❯s❡ ♣❛rts ✭❜✮ ❛♥❞ ✭❝✮ t♦ ❞❡r✐✈❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ❛♥❞ ❝♦♥❝❧✉❞❡ t❤❛t ❡✈❡r② x ∈ Rn ✐s %ˆ✲

❝♦♠♣❛r❛❜❧❡ t♦ 0✳ ❈♦♥❝❧✉❞❡ t❤❛t %ˆ ✐s ❝♦♠♣❧❡t❡✳

❙♦❧✉t✐♦♥✳

✭❛✮ ❉❡✜♥❡ A ❛s t❤❡ s❡t ♦❢ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ♣r❡♦r❞❡rs ♦♥ Rn t❤❛t ❡①t❡♥❞ %✳ ▲❡t _{B ⊆ A}

❜❡ s✉❝❤ t❤❛t (B,⊇) ✐s ❛ ❧♦s❡t ✭t❤❛t ✐s✱ t❤❡ r❡❧❛t✐♦♥ ⊇ ✇❤❡♥ r❡str✐❝t❡❞ t♦ B ✐s ❝♦♠♣❧❡t❡✮✳ ❉❡✜♥❡_%¯ _{:=∪B✳ ▲❡t✬s ♥♦✇ s❤♦✇ t❤❛t%}¯ _{✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ❢♦rB ✇✐t❤ r❡s♣❡❝t t♦ ⊇✳ ❚❤❛t}

✐s✱ ❧❡t✬s ♥♦✇ s❤♦✇ t❤❛t_%¯ _{✐s ❛ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ♣r❡♦r❞❡r t❤❛t ❡①t❡♥❞s%✳ ❙✉♣♣♦s❡ ✜rst}

t❤❛t x, y _∈ Rn _{❛r❡ s✉❝❤ t❤❛t x%}¯_{y ❛♥❞ ♣✐❝❦ ❛♥② z ∈ R}n_{✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ❡①✐sts} ˜

% _{∈ B} s✉❝❤ t❤❛t x%˜y✳ ❙✐♥❝❡ %˜ ✐s tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t✱ ✇❡ ❤❛✈❡ t❤❛t x+z%˜y+z ❛♥❞✱

❝♦♥s❡q✉❡♥t❧②✱x+z%¯y+z✳ ❚❤❛t ✐s✱%¯ ✐s tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t✳ ❙✐♥❝❡ ❛❧❧ r❡❧❛t✐♦♥s ✐♥_B ❛r❡

r❡✢❡①✐✈❡✱ ✐t ✐s ❝❧❡❛r t❤❛t_%¯ _{✐s ❛❧s♦ r❡✢❡①✐✈❡✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛tx, y, z ∈ R}n _{❛r❡ s✉❝❤ t❤❛t} x%¯y ❛♥❞ y%¯z✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ❡①✐st %1,%2∈ B ✇✐t❤ x %1 y ❛♥❞ y %2 z✳ ❙✐♥❝❡ ⊇

✐s ❛ ❧✐♥❡❛r ♦r❞❡r ♦♥ B✱ ✇❡ ❤❛✈❡ %1⊆%2 ♦r %2⊆%1✳ ❚❤✐s ♥♦✇ ✐♠♣❧✐❡s t❤❛t ❢♦r i= 1 ♦r 2✱

✇❡ ❤❛✈❡x%i y ❛♥❞ y %i z✳ ❙✐♥❝❡ %i ✐s tr❛♥s✐t✐✈❡✱ ✇❡ ❣❡t x%i z ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ x%¯z✳

❲❡ ❧❡❛r♥ t❤❛t _%¯ _{✐s tr❛♥s✐t✐✈❡✳ ■t ✐s ❛❧s♦ ❝❧❡❛r t❤❛t✱ ❢♦r ❛♥② x, y ∈R}n_{✱ ✐❢ x%y✱ t❤❡♥ x%}¯_y✳

❙✉♣♣♦s❡ ♥♦✇ t❤❛tx_≻y✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t ❢♦r ❛❧❧ r❡❧❛t✐♦♥s%˜ _{∈ B} ✇❡ ❛❧s♦ ❤❛✈❡x_≻˜y✳ ■t ✐s

♥♦✇ ❝❧❡❛r t❤❛t ✇❡ ♠✉st ❤❛✈❡ x_≻¯y✳ ❚❤✐s s❤♦✇s t❤❛t %¯ ✐s ❛ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ♣r❡♦r❞❡r

t❤❛t ❡①t❡♥❞s%✳ ●✐✈❡♥ t❤❡ ❛r❜✐tr❛r✐♥❡ss ♦❢ t❤❡ s❡tB✱ ✇❡ ❝❛♥ ♥♦✇ ❛♣♣❧② t❤❡ ❩♦r♥✬s ❧❡♠♠❛ ✷_{❋♦r♠❛❧❧②✱ ✇❡ ♠❡❛♥ t❤❛t t❤❡ s❡tY t♦❣❡t❤❡r ✇✐t❤ t❤❡ r❡str✐❝t✐♦♥ ♦❢%t♦ t❤❡ s❡tY ✭%∩(Y ×Y)✮ ❝♦♥st✐t✉t❡s ❛}

♣r❡♦r❞❡r❡❞ s❡t✳

✸_{❖t❤❡r✇✐s❡ ✇❡ ✇♦✉❧❞ ❤❛✈❡x≻y≻x}∗✱ ❝♦♥tr❛❞✐❝t✐♥❣ t❤❡ ❢❛❝t t❤❛t_x∗_∈▼❆❳_(Y,%)✳

✹_{◆♦t❛t✐♦♥✿ ❋♦r ❛♥② t✇♦ s❡ts A ❛♥❞ B✱ ✇❡ ✇r✐t❡ A−B t♦ r❡♣r❡s❡♥t t❤❡ s❡t C ❣✐✈❡♥ ❜② C := {x−y : x ∈}

✶✳ ❇■◆❆❘❨ ❘❊▲❆❚■❖◆❙ ✻ t♦ ✜♥❞ ❛ ♠❛①✐♠❛❧ ✭✇✐t❤ r❡s♣❡❝t t♦ t❤❡ r❡❧❛t✐♦♥ ⊇✮ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ♣r❡♦r❞❡r _%ˆ _t❤❛t

❡①t❡♥❞s %✳

✭❜✮ ❋♦❧❧♦✇✐♥❣ t❤❡ st❡♣s ✐♥ t❤❡ ❡①❡r❝✐s❡✱ s✉♣♣♦s❡ t❤❛tx∈Rn✐s s✉❝❤ t❤❛tx✐s ♥♦t%ˆ✲❝♦♠♣❛r❛❜❧❡

t♦ 0✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❢♦r ♥♦ k ∈ N ✇❡ ❤❛✈❡ kx_≻ˆ0✳ ◆♦✇ ❞❡✜♥❡ t❤❡ r❡❧❛t✐♦♥ %∗ _❜②

y %∗ _{z ⇐⇒ y −z ∈ (0}↑,%ˆ _{− [[x]])✱ ❛s s✉❣❣❡st❡❞ ✐♥ t❤❡ q✉❡st✐♦♥✳ ❲❡ ✜rst ♥♦t❡ t❤❛t}

0₋x _∈ (0↑,%ˆ _−[[x]])✱ s♦ t❤❛t _0%∗ _x✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t _y%ˆ_z✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t _{y−z ∈} 0↑,%ˆ _⊆(0↑,%ˆ_−[[x]])❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱_y%∗ _z✳ ◆♦✇ s✉♣♣♦s❡ t❤❛t_y≻ˆ_z✳ ■❢_z−y∈(0↑,%ˆ_−[[x]])✱

t❤❡♥ t❤❡r❡ ❡①✐stw∈0↑,%ˆ ❛♥❞ _k∈Z+ s✉❝❤ t❤❛t _z−y=w−kx✳ ❚❤❛t ✐s✱_kx=w+y−z✳

❙✐♥❝❡ w%ˆ0 ❛♥❞ y₋z_≻ˆ0✱ t❤✐s ✐♠♣❧✐❡s t❤❛t kx_≻ˆ0✱ ✇❤✐❝❤ ✐s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❲❡ ❝♦♥❝❧✉❞❡

t❤❛t z₋y /_∈(0↑,%ˆ _−[[x]]) ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ _{y ≻}∗ _z✳ ❚❤✐s s❤♦✇s t❤❛t _%∗ ❡①t❡♥❞s _%ˆ ❛♥❞✱

❝♦♥s❡q✉❡♥t❧②✱ ✐t ❛❧s♦ ❡①t❡♥❞s%✳ ❆s ✐t ✐s r♦✉t✐♥❡ t♦ s❤♦✇ t❤❛t%∗ _{✐s ❛ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t}

♣r❡♦r❞❡r✱ ✇❡ ❛rr✐✈❡ ❛t ❛ ❝♦♥tr❛❞✐❝t✐♦♥ t♦ t❤❡ ♠❛①✐♠❛❧✐t② ♦❢_%ˆ_{✱ s✐♥❝❡0%}∗ _{x✱ ❜✉t ✇❡ ❞♦ ♥♦t}

❤❛✈❡0 ˆ%x✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡r❡ ♠✉st ❡①✐st k ∈N ✇✐t❤ kx_≻ˆ0✳

✭❝✮ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t ❢♦r ♥♦ k ∈ N ✇❡ ❤❛✈❡ 0 ˆ≻kx✳ ◆♦✇ ❞❡✜♥❡ %∗ _{❜② y %}∗ _{z ⇐⇒ y−z ∈}

(0↑,%ˆ _{+ [[x]])}✳ ◆♦t❡ t❤❛t _x−0∈(0↑,%ˆ _{+ [[x]])}✱ s♦ t❤❛t_x%∗ ₀✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t_y%ˆ_z✳ ❚❤✐s

✐♠♣❧✐❡s t❤❛ty₋z _∈0↑,%ˆ _⊆(0↑,%ˆ_{+ [[x]])}❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱_y%∗ _z✳ ◆♦✇ s✉♣♣♦s❡ t❤❛t_y≻ˆ_z✳

■❢ z−y ∈(0↑,%ˆ _{+ [[x]])}✱ t❤❡♥ t❤❡r❡ ❡①✐st_{w ∈} 0↑,%ˆ ❛♥❞ _{k ∈Z+} s✉❝❤ t❤❛t _{z−y =w+kx}✳

❚❤❛t ✐s✱ kx = z−y−w✳ ❙✐♥❝❡ 0 ˆ%₋w ❛♥❞ 0 ˆ≻z−y✱ t❤✐s ✐♠♣❧✐❡s t❤❛t 0 ˆ≻kx ✇❤✐❝❤ ✐s

❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t z−y /∈ (0↑,%ˆ _{+ [[x]])} ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ _{y ≻}∗ _z✳ ❚❤✐s

s❤♦✇s t❤❛t %∗ ❡①t❡♥❞s %ˆ ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ ✐t ❛❧s♦ ❡①t❡♥❞s %✳ ❆s ✐t ✐s r♦✉t✐♥❡ t♦ s❤♦✇

t❤❛t%∗ ✐s ❛ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ♣r❡♦r❞❡r✱ ✇❡ ❛rr✐✈❡ ❛t ❛ ❝♦♥tr❛❞✐❝t✐♦♥ t♦ t❤❡ ♠❛①✐♠❛❧✐t②

♦❢ _%ˆ_{✱ s✐♥❝❡ x %}∗ _{0✱ ❜✉t ✇❡ ❞♦ ♥♦t ❤❛✈❡ x%}ˆ_{0✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡r❡ ♠✉st ❡①✐st k ∈ N}

✇✐t❤ 0 ˆ≻kx✳

✭❞✮ ❇② ♣❛rts ✭❜✮ ❛♥❞ ✭❝✮✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡r❡ ❡①✐stsk1 ❛♥❞k2✐♥Ns✉❝❤ t❤❛tk1x≻ˆ0❛♥❞0 ˆ≻k2x✳

❙✐♥❝❡_%ˆ _{✐s ❛ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t ♣r❡♦r❞❡r✱ t❤✐s ✐♠♣❧✐❡s t❤❛tk}

1k2x≻ˆ0❛♥❞0 ˆ≻k1k2x✱ ✇❤✐❝❤

✐s ❛ ❝❧❡❛r ❝♦♥tr❛❞✐❝t✐♦♥✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❡✈❡r② x∈Rn ✐s %ˆ✲❝♦♠♣❛r❛❜❧❡ t♦ 0✳ ❇✉t t❤❡♥✱

❢♦r ❡✈❡r② x, y _∈ Rn _{✇❡ ❤❛✈❡ t❤❛t x−y%}ˆ_{0 ♦r 0 ˆ%x−y✳ ❙✐♥❝❡ %}ˆ _{✐s tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t✱}

✐♥ t❤❡ ✜rst ❝❛s❡ ✇❡ ❤❛✈❡ x%ˆy ❛♥❞ ✐♥ t❤❡ s❡❝♦♥❞ y%ˆx✳ ❚❤✐s s❤♦✇s t❤❛t %ˆ ✐s ❛ ❝♦♠♣❧❡t❡

❈❍❆P❚❊❘ ✷

❘❡✈❡❛❧❡❞ Pr❡❢❡r❡♥❝❡ ❚❤❡♦r②

❊①❡r❝✐s❡ ✷✳✶✳ ❙✉♣♣♦s❡ t❤❛t(X,ΩX)✐s ❛ ❝❤♦✐❝❡ s♣❛❝❡ ❛♥❞ t❤❛t c: ΩX _→2X

\ {∅} ✐s ❛ ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡✳ ❚❤❛t ✐s✱ ❢♦r ❛♥② A∈ΩX✱c(A)⊆A✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ c:

Pr♦♣❡rt② α✳ ❋♦r ❛♥②x∈X✱ ✐❢A, B ∈ΩX ❛r❡ s✉❝❤ t❤❛tB ⊆A❛♥❞x∈c(A)∩B✱ t❤❡♥x∈c(B)✳

Pr♦♣❡rt② β✳ ❋♦r ❛♥② x, y ∈ X✱ ✐❢ A, B ∈ ΩX ❛r❡ s✉❝❤ t❤❛t A ⊆ B✱ x, y ∈ c(A) ❛♥❞ y ∈ c(B)✱

t❤❡♥ x∈c(B).

❙❤♦✇ t❤❛t α ❛♥❞ β ❛r❡ ❡q✉✐✈❛❧❡♥t t♦ ❲❆❘P✳ ❚❤❛t ✐s✱ s❤♦✇ t❤❛t c s❛t✐s✜❡s α ❛♥❞ β ✐❢ ❛♥❞ ♦♥❧② ✐❢ cs❛t✐s✜❡s ❲❆❘P✳

❙♦❧✉t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛tcs❛t✐s✜❡s ❲❆❘P✳ P✐❝❦A, B ∈ΩX s✉❝❤ t❤❛tB ⊆A❛♥❞x∈c(A)∩B✳

◆♦✇ ♣✐❝❦ ❛♥② y ∈ c(B)✳ ■❢ x = y t❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ ❞♦✳ ■❢ x 6= y✱ t❤❡♥✱ s✐♥❝❡ y ∈ B ⊆ A✱ ❜②

❲❆❘P ✇❡ ♠✉st ❤❛✈❡ x∈c(B)✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛tcs❛t✐s✜❡sα✳ ◆♦✇ ♣✐❝❦x, y ∈X ❛♥❞A, B ∈ΩX

s✉❝❤ t❤❛t A ⊆ B✱ x, y ∈ c(A) ❛♥❞ y ∈ c(B)✳ ❙✐♥❝❡ x ∈ A ⊆ B✱ ❲❆❘P ✐♠❡❞✐❛t❡❧② ✐♠♣❧✐❡s t❤❛t x∈c(B)✳ ❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ t❤❛tcs❛t✐s✜❡sα❛♥❞β✳ ◆♦✇ s✉♣♣♦s❡ t❤❛tx, y ∈X ❛♥❞A, B ∈ΩX

❛r❡ s✉❝❤ t❤❛t x∈c(A)∩B ❛♥❞ y∈c(B)∩A✳ ❇② α✱ {x, y} ⊆c({x, y})✳ ❇✉t ♥♦✇ β ✐♠♣❧❡s t❤❛t

x_∈c(B)✳

❊①❡r❝✐s❡ ✷✳✷ ✭❇❛s❡❞ ♦♥ ▼❛♥❞❧❡r ❡t ❛❧✳ ✭✷✵✶✷✮✮✳ ▲❡t X ❜❡ ❛ ✜♥✐t❡ s❡t ♦❢ ❛❧t❡r♥❛t✐✈❡s✳ ❆

❝❤❡❝❦❧✐st ✐s ❛ ✜♥✐t❡ ❧✐st ♦❢ ♣r♦♣❡rt✐❡s t❤❛t ❡❛❝❤ ❛❧t❡r♥❛t✐✈❡ ♠✐❣❤t ♣♦ss❡s ♦r ♥♦t✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢

X ✐s ❛ s❡t ♦❢ ❝❛rs✱ t❤❡ ✜rst ✐t❡♠ ✐♥ t❤❡ ❧✐st ❝♦✉❧❞ ❜❡ ✐❢ t❤❡ ❝❛r ❤❛s ✹ ❞♦♦rs ♦r ♥♦t✳ ❚❤❡ s❡❝♦♥❞

❝♦✉❧❞ ❜❡ ✐❢ t❤❡ ♣r✐❝❡ ♦❢ t❤❡ ❝❛r ✐s ❜❡❧❧♦✇ ✺✵✳✵✵✵ r❡❛✐s ♦r ♥♦t✱ ❡t❝✳✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤♦✐❝❡ ♣r♦❝❡❞✉r❡✿ ●✐✈❡♥ ❛ s❡t ♦❢ ❛❧t❡r♥❛t✐✈❡s A✱ t❤❡ ❛❣❡♥t ✜rst s❡❡s ✐❢ s♦♠❡ ❛❧t❡r♥❛t✐✈❡ ✐♥ A s❛t✐s✜❡s t❤❡

✜rst ♣r♦♣❡rt② ✐♥ t❤❡ ❝❤❡❝❦❧✐st✳ ■❢ t❤✐s ✐s tr✉❡✱ t❤❡♥ t❤❡ ❛❣❡♥t ❡❧✐♠✐♥❛t❡s ❛❧❧ t❤❡ ❛❧t❡r♥❛t✐✈❡s ✐♥At❤❛t

❞♦ ♥♦t s❛t✐s❢② t❤❡ ✜rst ♣r♦♣❡rt② ❛♥❞ ♠♦✈❡s ♦♥ t♦ t❤❡ ♥❡①t ✐t❡♠ ✐♥ t❤❡ ❝❤❡❝❦❧✐st✳ ■❢ ♥♦ ❛❧t❡r♥❛t✐✈❡ ✐♥A s❛t✐s✜❡s t❤❡ ✜rst ♣r♦♣❡rt②✱ t❤❡♥ t❤❡ ❛❣❡♥t ♠♦✈❡s ♦♥ t♦ t❤❡ ♥❡①t ✐t❡♠ ✐♥ t❤❡ ❝❤❡❝❦❧✐st ✇✐t❤♦✉t

❡❧✐♠✐♥❛t✐♥❣ ❛♥② ❛❧t❡r♥❛t✐✈❡✳ ❘❡♣❡❛t t❤✐s ♣r♦❝❡❞✉r❡ ✉♥t✐❧ t❤❡ ❡♥❞ ♦❢ t❤❡ ❝❤❡❝❦ ❧✐st✳ ❚❤❡ ❛❧t❡r♥❛t✐✈❡s t❤❛t s✉r✈✐✈❡ t❤❡ ♣r♦❝❡❞✉r❡ ✉♥t✐❧ t❤❡ ❡♥❞ ❛r❡ t❤❡ ❛❣❡♥t✬s ❝❤♦✐❝❡✳

✭❛✮ ❙❤♦✇ t❤❛t ✐❢ t❤❡ ❛❣❡♥t ♠❛❦❡s ❤❡r ❝❤♦✐❝❡s ❛❝❝♦r❞✐♥❣ t♦ ❛ ❝❤❡❝❦❧✐st✱ t❤❡♥ ❤❡r ❝❤♦✐❝❡s ❣❡♥❡r❛t❡ ❛ ✇❡❧❧✲❞❡✜♥❡❞ ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❛♥❞ t❤✐s ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ s❛t✐s✜❡s ❲❆❘P✳ ✭❜✮ ❙❤♦✇ t❤❛t t❤❡ ❝♦♥✈❡rs❡ ✐s ❛❧s♦ tr✉❡✳ ❚❤❛t ✐s✱ ✐❢c✐s ❛ ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♦♥ ❛ ✜♥✐t❡ s❡t

X ❛♥❞ c s❛t✐s✜❡s ❲❆❘P✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝❤❡❝❦❧✐st t❤❛t ❣❡♥❡r❛t❡s c✳ ✭❍✐♥t✿ ❯s❡ t❤❡

s✉❜s❡ts str✉❝t✉r❡ ♦❢X t♦ ❞♦ t❤❛t✱ ❣♦✐♥❣ ❢r♦♠ t❤❡ ❧❛r❣❡st s✉❜s❡t t♦ t❤❡ s♠❛❧❧❡st✳ ❍❛✈❡ ♥♦

s❤❛♠❡✱ t❤❡ ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❝❤❡❝❦❧✐st ❝❛♥ ❜❡ ❛♥②t❤✐♥❣ ②♦✉ ✇❛♥t✳✮

✭❝✮ ❙♦✱ ✐❢ X ✐s ✜♥✐t❡✱ ❜② s♦♠❡ r❡s✉❧ts ✇❡ ❤❛✈❡ st✉❞✐❡❞ ❜❡❢♦r❡✱ t❤✐s ✐♠♣❧✐❡s t❤❛t ❛ ❝❤♦✐❝❡ ❝♦r✲

r❡s♣♦♥❞❡♥❝❡ ❤❛s ❛ r❡♣r❡s❡♥t❛t✐♦♥ ❜② ❛ ❝❤❡❝❦❧✐st ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤✐s ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♠❛①✐♠✐③❡s s♦♠❡ ❝♦♠♣❧❡t❡ ♣r❡❢❡r❡♥❝❡✱ ✇❤✐❝❤ ❤❛♣♣❡♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤✐s ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥✲ ❞❡♥❝❡ ♠❛①✐♠✐③❡s s♦♠❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥✳ ❚❤✐s s❤♦✉❧❞ ♥♦t ❜❡ ❛ s✉r♣r✐s❡ ❛t ❛❧❧✳ ❲❤②❄ ✭❍✐♥t✿ ❚❤❡ ❛♥s✇❡r ❝♦♠❡s ❢r♦♠ ❛ s✐♠♣❧❡ ♠❛t❤❡♠❛t✐❝❛❧ ♦❜s❡r✈❛t✐♦♥✱ s♦ ♣❧❡❛s❡ ❞♦ ♥♦t ❡❧❛❜♦r❛t❡ ♦♥ ②♦✉r ❛r❣✉♠❡♥t✳ ❨♦✉ ♠❛② s❡❡ t❤❡ ♦❜s❡r✈❛t✐♦♥ ♦r ♥♦t✱ ❜✉t✱ ✐♥ ❜♦t❤ ❝❛s❡s✱ ②♦✉r ❛♥s✇❡r s❤♦✉❧❞ ❜❡ ✈❡r② s❤♦rt✳✮

✷✳ ❘❊❱❊❆▲❊❉ P❘❊❋❊❘❊◆❈❊ ❚❍❊❖❘❨ ✽ ❙♦❧✉t✐♦♥✳

✭❛✮ ■t ✐s ❝❧❡❛r t❤❛t ✐♥ ❡✈❡r② st❡♣ ♦❢ t❤❡ ❝❤❡❝❦❧✐st ♣r♦❝❡❞✉r❡ t❤❡r❡ ✐s s♦♠❡ ❛❧t❡r♥❛t✐✈❡ t❤❛t s✉r✈✐✈❡s t❤❡ st❡♣✳ ❙✐♥❝❡ t❤❡ ♥✉♠❜❡r ♦❢ st❡♣s ✐s ✜♥✐t❡✱ t❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ✐s ❛❧✇❛②s s♦♠❡ ❛❧t❡r♥❛t✐✈❡ t❤❛t s✉r✈✐✈❡s t❤❡ ❡♥t✐r❡ ♣r♦❝❡ss ❛♥❞✱ t❤❡r❡❢♦r❡✱ t❤❡ ♣r♦❝❡ss ✐♥❞✉❝❡s ❛ ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡✳ ▲❡t c ❜❡ t❤❡ ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✐♥❞✉❝❡❞ ❜② s♦♠❡ ❝❤❡❝❦❧✐st✳

❙✉♣♣♦s❡ t❤❛tx, y ∈X ❛♥❞ A, B ⊆X ❛r❡ s✉❝❤ t❤❛tx∈c(A)∩B ❛♥❞ y∈c(B)∩A✳ ■t ✐s

❡❛s② t♦ s❡❡ t❤❛t t❤✐s ❝❛♥ ♦♥❧② ❤❛♣♣❡♥ ✐❢x ❛♥❞ y ❛❣r❡❡ ❢♦r ❛❧❧ ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❝❤❡❝❦ ❧✐st✳

❚❤❛t ✐s✱ ❢♦r ❡❛❝❤ ♣r♦♣❡rt②i✐♥ t❤❡ ❝❤❡❝❦❧✐st✱ x❤❛s ♣r♦♣❡rt②i ✐❢ ❛♥❞ ♦♥❧② ✐❢y❤❛s ♣r♦♣❡rt② i✳ ❇✉t t❤❡♥ ✐t ✐s ❝❧❡❛r t❤❛t y _∈c(B) ✐♠♣❧✐❡s t❤❛t x _∈c(B)✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t cs❛t✐s✜❡s

❲❆❘P✳

✭❜✮ ❉❡✜♥❡✱ ✐♥❞✉❝t✐✈❡❧②✱ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ ♦❢ s❡ts✳ A0 := X ❛♥❞✱ ❢♦r ❡❛❝❤ i ≥ 1✱ Ai :=

Ai−1\c(Ai−1)✳ ▲❡tI ∈N❜❡ t❤❡ ✜rst ♥✉♠❜❡r s✉❝❤ t❤❛tAI+1 =∅✳ ❋♦r ❡❛❝❤i= 1, ..., I ❛♥❞

x_∈X✱ ❧❡t t❤❡ ♣r♦♣❡rt② i ✐♥ t❤❡ ❝❤❡❝❦❧✐st ❜❡ ✐❢ x_∈c(Ai)✳ ◆♦✇ ❧❡tB ❜❡ ❛♥② s✉❜s❡t ♦❢ X✳

❚❤❡r❡ ♠✉st ❡①✐st s♦♠❡ ♠❛①✐♠✉♠ i∗ _{∈ {0, ..., I} s✉❝❤ t❤❛t B ⊆ Ai∗✳ ◆♦t❡ t❤❛t✱ ❢♦r ❡❛❝❤}

i < i∗_{✱ t❤❡ ❛♥s✇❡r ❢♦r t❤❡ ith♣r♦♣❡rt② ✐s ♥♦ ❢♦r ❛❧❧x∈B✳ ❆❧s♦✱ ♥♦t❡ t❤❛t✱ ❜② ❝♦♥str✉❝t✐♦♥✱}

c(Ai∗)∩B 6=∅✳ ❙✐♥❝❡ B ⊆ Ai∗✱ ❲❆❘P ✐♠♣❧✐❡s t❤❛t c(B) = c(Ai∗)∩B✳ ❇✉t t❤❡s❡ ❛r❡

❡①❛❝t❧② t❤❡ ❡❧❡♠❡♥ts ♦❢ B ✇❤♦s❡ ❛♥s✇❡r t♦ t❤❡ i∗

th ♣r♦♣❡rt② ✐s ②❡s✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡

♣r♦♣♦s❡❞ ❝❤❡❝❦❧✐st ✐♥❞❡❡❞ r❡♣r❡s❡♥ts c✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ♦t❤❡r ❞✐✛❡r❡♥t ❝❤❡❝❦❧✐sts t❤❛t

❝♦✉❧❞ r❡♣r❡s❡♥t t❤❡ s❛♠❡ ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡✳ ❈❛♥ ②♦✉ t❤✐♥❦ ♦❢ ❛ ❝❤❡❝❦❧✐st ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♦♥❡ ♣r❡s❡♥t❡❞ ❤❡r❡❄

✭❝✮ ❙✐♥❝❡ X ✐s ✜♥✐t❡ ❛♥❞ c s❛t✐s✜❡s ❲❆❘P✱ ✇❡ ❦♥♦✇ t❤❛t c♠❛①✐♠✐③❡s s♦♠❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥✳

▼♦r❡♦✈❡r✱ ✐t ✐s ♥♦t ❤❛r❞ t♦ s❡❡ t❤❛t ✇❡ ❝♦✉❧❞ r❡♣r❡s❡♥t c ✉s✐♥❣ ❛ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ ✇❤♦s❡

✐♠❛❣❡ ✐♥❝❧✉❞❡s ♦♥❧② ♥❛t✉r❛❧ ♥✉♠❜❡rs✳ ❊✈❡r② ♥❛t✉r❛❧ ♥✉♠❜❡r ❤❛s ❛ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ ❜✐♥❛r② ❜❛s❡ ✇✐t❤ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❞✐❣✐ts✳ ❇✉t ✐❢ ②♦✉ ❧♦♦❦ ❝❧♦s❡❧②✱ ❛ ❝❤❡❝❦❧✐st ✐s ♥♦t❤✐♥❣ ♠♦r❡ t❤❛♥ ❛ ❜✐♥❛r② ❜❛s❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ t❤❛t r❡t✉r♥s ♦♥❧② ♥❛t✉r❛❧ ♥✉♠❜❡rs✳ ❙♦✱ ❞❡s♣✐t❡ ❛❧❧ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❝❤♦✐❝❡ ❜② ❝❤❡❝❦❧✐st ♣r♦❝❡❞✉r❡✱ ❢♦r♠❛❧❧②✱ ✐t ✐s t❤❡ s❛♠❡ t❤✐♥❣ ❛s r❡♣r❡s❡♥t❛t✐♦♥ ❜② ❛ ✉t✐❧✐t② ❢✉♥❝t✐♦♥✳ ◆♦t❡✿ ❋♦r s✐♠♣❧✐❝✐t②✱ ■ ❛ss✉♠❡❞ t❤❛t

X ✇❛s ✜♥✐t❡ ✐♥ t❤✐s ❡①❡r❝✐s❡✱ ❜✉t t❤❡ s❛♠❡ ❝♦♥❝❧✉s✐♦♥s ✇♦✉❧❞ st✐❧❧ ❜❡ tr✉❡ ✐❢ ✇❡ ❛❧❧♦✇❡❞

❢♦r t❤❡ ✉s❡ ♦❢ ❝❤❡❝❦❧✐sts ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡ ♥✉♠❜❡r ♦❢ ♣r♦♣❡rt✐❡s✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t ❛♥② r❡❛❧ ♥✉♠❜❡r ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ s❡q✉❡♥❝❡ ♦❢ ③❡r♦s ❛♥❞ ♦♥❡s✳

❊①❡r❝✐s❡ ✷✳✸✳ ❙✉♣♣♦s❡ t❤❛t(X,ΩX) ✐s ❛ ❝❤♦✐❝❡ s♣❛❝❡ ❛♥❞ t❤❛tc: ΩX _→2X _{\ {∅} ✐s ❛ ❝❤♦✐❝❡}

❝♦rr❡s♣♦♥❞❡♥❝❡✳ ❚❤❛t ✐s✱ ❢♦r ❛♥② A_∈ΩX✱c(A)_⊆A✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt② ♦❢ c:

Pr♦♣❡rt② γ✳ ❙✉♣♣♦s❡ S_∈ΩX ✐s s✉❝❤ t❤❛t S =S

{T :T _{∈ M}} ❢♦r s♦♠❡ ♥♦♥❡♠♣t② _{M ⊆} ΩX✳ ■❢ x_∈c(T)_∀T _{∈ M}✱ t❤❡♥ x_∈c(S)✳

❙❤♦✇ t❤❛t α ❛♥❞ γ ❛r❡ ❡q✉✐✈❛❧❡♥t t♦ ❲❲❆❘◆■✳ ❚❤❛t ✐s✱ s❤♦✇ t❤❛tcs❛t✐s✜❡s α ❛♥❞ γ ✐❢ ❛♥❞ ♦♥❧②

✐❢ c s❛t✐s✜❡s ❲❲❆❘◆■✳

❙♦❧✉t✐♦♥✳ ■t ✐s ❡❛s② t♦ ❝❤❡❝❦ t❤❛t ❲❲❆❘◆■ ✐♠♣❧✐❡s α ❛♥❞ γ✳ ❙✉♣♣♦s❡ t❤❛t α ❛♥❞ γ ❤♦❧❞✳

▲❡t S _∈ ΩX ❛♥❞ x _∈ S✳ ▼♦r❡♦✈❡r✱ ❛ss✉♠❡ t❤❛t ❢♦r ❡✈❡r② y _∈ S t❤❡r❡ ❡①✐sts ❛ Ty _∈ ΩX ✇✐t❤ x_∈c(Ty)❛♥❞ y_∈Ty✳ ❙✐♥❝❡ _{x, y_{} ⊆}Ty ❛♥❞x_∈c(Ty)✱ t❤❡α✲❛①✐♦♠ ❣✉❛r❛♥t❡❡s t❤❛t x_∈c(_{x, y_})✱

❢♦r ❡✈❡r② y_∈S✳ ❇② t❤❡ γ✲❛①✐♦♠✱ x_∈cS

y∈S{x, y}

=c(S)✱ ❛s ❞❡s✐r❡❞✳

❊①❡r❝✐s❡ ✷✳✹ ✭❇❛s❡❞ ♦♥ ❙❡♥ ✭✶✾✼✶✮✮✳ ▲❡t (X,ΩX) ❜❡ ❛♥② ❝❤♦✐❝❡ s♣❛❝❡✳ ❲❡ s❛② t❤❛t ❛ ❝❤♦✐❝❡

❝♦rr❡s♣♦♥❞❡♥❝❡ c ♦♥ (X,ΩX) ✐s r❡❣✉❧❛r ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝♦♠♣❧❡t❡ ✭❜✉t ♣❡r❤❛♣s ♥♦t tr❛♥s✐t✐✈❡✮

❜✐♥❛r② r❡❧❛t✐♦♥ %_⊆ X×X s✉❝❤ t❤❛t c(S) = max(S,%) ❢♦r ❡✈❡r② S ∈ ΩX✳ ❙❤♦✇ t❤❛t ❛ ❝❤♦✐❝❡

✷✳ ❘❊❱❊❆▲❊❉ P❘❊❋❊❘❊◆❈❊ ❚❍❊❖❘❨ ✾ ❙♦❧✉t✐♦♥✳ ■t ✐s ❝❧❡❛r t❤❛t ✐❢ c ✐s r❡❣✉❧❛r✱ t❤❡♥ ✐t s❛t✐s✜❡s α ❛♥❞ γ✱ s♦ ✇❡ ✇✐❧❧ ♦♥❧② s❤♦✇ t❤❡

❝♦♥✈❡rs❡✳ ❙✉♣♣♦s❡✱ t❤❡♥✱ t❤❛t c s❛t✐s✜❡s α ❛♥❞ γ✳ ❉❡✜♥❡ t❤❡ ❜✐♥❛r② r❡❧❛t✐♦♥ %_⊆ X _×X ❜② x%y _⇐⇒ x_∈c(_{x, y_})✳ ■t ✐s ❝❧❡❛r t❤❛t %✐s ❝♦♠♣❧❡t❡✳ ◆♦✇ ✜① ❛♥② ❝❤♦✐❝❡ ♣r♦❜❧❡♠ S_∈ΩX ❛♥❞

s✉♣♣♦s❡ t❤❛t x_∈c(S)✳ ❇② α✱ ✇❡ ❤❛✈❡ t❤❛t x_∈ c(_{x, y_}) ❢♦r ❡✈❡r② y_∈S, s♦ t❤❛t x _∈max(S,%)✳

❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ t❤❛t x _∈max(S,%)✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t x _∈c(_{x, y_}) ❢♦r ❡✈❡r② y _∈S✳ ◆♦✇ γ

✐♠♣❧✐❡s t❤❛t x_∈c(S

y∈S{x, y}) = c(S)✳

❊①❡r❝✐s❡ ✷✳✺ ✭❇❛s❡❞ ♦♥ ❑r❡♣s ✭✶✾✼✾✮✮✳ ▲❡t X ❜❡ ❛ ✜♥✐t❡ s❡t ♦❢ ❛❧t❡r♥❛t✐✈❡s ❛♥❞ ❞❡✜♥❡ _A:= 2X _{\ {∅}✳ ❲❡ ❝❛❧❧ t❤❡ ❡❧❡♠❡♥ts ♦❢ A ♠❡♥✉s ❛♥❞ ♦✉r ♦❜❥❡❝t ♦❢ st✉❞② ✇✐❧❧ ❜❡ ❛ ❜✐♥❛r② r❡❧❛t✐♦♥}

%_{⊆ A × A}. ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t②♣❡ ♦❢ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r %✳ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♠♣❧❡t❡

♣r❡♦r❞❡r %∗_⊆X_×X s✉❝❤ t❤❛t✱ ❢♦r ❛♥② t✇♦ ♠❡♥✉s A ❛♥❞ B✱A%B ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❛♥

❡❧❡♠❡♥t x_∈A s✉❝❤ t❤❛tx%y ❢♦r ❡✈❡r② y_∈B✳

✭❛✮ ❙❤♦✇ t❤❛t ✐❢ % ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛❜♦✈❡✱ t❤❡♥% ✐s ❛ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡r✳

✭❜✮ ❈❛♥ ②♦✉ ❛①✐♦♠❛t✐③❡ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❛❜♦✈❡❄ ❨♦✉ ❝❛♥ ❞♦ t❤❛t ✇✐t❤ ❛ s✐♥❣❧❡ s✐♠♣❧❡ ❛①✐♦♠ ✭❛❢t❡r ②♦✉ ✐♠♣♦s❡ t❤❛t % ✐s ❛ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡r✮✱ ❜✉t ❛♥②✇❛② ②♦✉ ❞♦ t❤❛t ✇✐❧❧ ❜❡

✐♥t❡r❡st✐♥❣✳ ■❢ ②♦✉ ❝❛♥♥♦t ❝♦♠❡ ✉♣ ✇✐t❤ t❤❡ ❛①✐♦♠ ②♦✉ ❝❛♥ ✜♥❞ ✐t ✐♥ ❑r❡♣s ✭✶✾✼✾✮✳ ❙♦❧✉t✐♦♥✳

✭❛✮ ❋✐① ❛♥② t✇♦ ♠❡♥✉s A ❛♥❞ B✳ ❙✐♥❝❡ %∗ ✐s ❛ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡r✱ t❤❡r❡ ❡①✐sts x _∈ A_∪B

s✉❝❤ t❤❛t x %∗ y ❢♦r ❛❧❧ y _∈ A_∪B✳ ■t ✐s ❝❧❡❛r t❤❛t ✐❢ x_∈ A✱ A % B ❛♥❞ ✐❢ x_∈ B✱ t❤❡♥ B %A✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t % ✐s ❝♦♠♣❧❡t❡✳ ◆♦✇ s✉♣♣♦s❡ A, B ❛♥❞ C ❛r❡ s✉❝❤ t❤❛t A%B

❛♥❞ B %C✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ❡①✐sts xA _∈ A s✉❝❤ t❤❛t xA %∗ _{y ❢♦r ❛❧❧ y ∈ B ❛♥❞}

t❤❡r❡ ❡①✐sts xB _∈B s✉❝❤ t❤❛t xB %∗ _{y ❢♦r ❛❧❧ y ∈C✳ ■♥ ♣❛rt✐❝✉❧❛r✱ xA%}∗ _{xB✱ ✇❤✐❝❤✱ ❜②}

t❤❡ tr❛♥s✐t✐✈✐t② ♦❢ %∗_{✱ ✐♠♣❧✐❡s t❤❛t xA%}∗ _{y ❢♦r ❛❧❧ y ∈ C✳ ❚❤❛t ✐s✱ A % C✳ ❲❡ ❝♦♥❝❧✉❞❡}

t❤❛t %✐s tr❛♥s✐t✐✈❡✳ ■t ✐s ❝❧❡❛r t❤❛t %✐s ❛❧s♦ r❡✢❡①✐✈❡✱ s♦ t❤❛t ✐t ✐s ❛ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡r✳

✭❜✮ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦st✉❧❛t❡✿

❆①✐♦♠ ✭❙❡t ❘❛t✐♦♥❛❧✐t②✮✳ ❋♦r ❛♥② t✇♦ ♠❡♥✉s A ❛♥❞ B✱ ✐❢ A%B✱ t❤❡♥ A∼A∪B✳

❲❡ ❝❛♥ ♥♦✇ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✿

❚❤❡♦r❡♠ ✷✳✶✳ % ✐s ❛ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡r t❤❛t s❛t✐s✜❡s ❙❡t ❘❛t✐♦♥❛❧✐t② ✐❢ ❛♥❞ ♦♥❧② ✐❢

t❤❡r❡ ❡①✐sts ❛ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡r %∗ _{s✉❝❤ t❤❛t✱ ❢♦r ❛♥② t✇♦ ♠❡♥✉s A ❛♥❞ B✱ A%B ⇐⇒}

t❤❡r❡ ❡①✐sts x_∈A ✇✐t❤ x%∗ _{y ❢♦r ❛❧❧ y∈B.}

❋♦r t❤❛t✱ s✉♣♣♦s❡ ✜rst t❤❛t % ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ✐♥ t❤❡ st❛t❡♠❡♥t ♦❢ t❤❡ t❤❡♦r❡♠

❢♦r s♦♠❡ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡r %∗_{✳ ■♥ ♣❛rt ✭❛✮ ✇❡ ❤❛✈❡ ❛❧r❡❛❞② s❤♦✇♥ t❤❛t % ✐s ❛ ❝♦♠♣❧❡t❡}

♣r❡♦r❞❡r✳ ❈♦♥s✐❞❡r ♥♦✇ t✇♦ ♠❡♥✉s A ❛♥❞ B s✉❝❤ t❤❛t A % B✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡r❡

❡①✐stsx_∈A s✉❝❤ t❤❛tx%∗ y ❢♦r ❛❧❧ y_∈B✳ ❙✐♥❝❡ %∗ ✐s ❛ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡r✱ t❤❡r❡ ❡①✐sts

x∗ _{∈ A s✉❝❤ t❤❛t x}∗ _%∗ _{y ❢♦r ❛❧❧ y ∈ A✳ ■♥ ♣❛rt✐❝✉❧❛r✱ x}∗ _%∗ _{x✱ s♦✱ ❜② t❤❡ tr❛♥s✐t✐✈✐t②}

♦❢ %∗✱ ✇❡ ❤❛✈❡ t❤❛t x∗ _%∗ _{y ❢♦r ❛❧❧ y ∈ A ∪B✳ ❚❤❛t ✐s✱ A % A ∪B✳ ❙✐♥❝❡ x}∗ _✐s

❛❧s♦ ❛♥ ❡❧❡♠❡♥t ♦❢ A _∪B✱ ✇❡ ❛❧s♦ ❤❛✈❡ A_∪B % A✱ s♦ t❤❛t A _∼ A_∪B✳ ❈♦♥✈❡rs❡❧②✱

s✉♣♣♦s❡ t❤❛t % ✐s ❛ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡r t❤❛t s❛t✐s✜❡s ❙❡t ❘❛t✐♦♥❛❧✐t②✳ ❉❡✜♥❡ t❤❡ ❜✐♥❛r②

r❡❧❛t✐♦♥ %∗ _{❜② x %}∗ _{y ⇐⇒ {x} % {y}✱ ❢♦r ❛♥② x, y ∈ X✳ ❙✐♥❝❡ % ✐s ❛ ❝♦♠♣❧❡t❡}

♣r❡♦r❞❡r ❛♥❞ %∗ _{❝❛♥ ❜❡ s❡❡♥ ❛s t❤❡ r❡str✐❝t✐♦♥ ♦❢% t♦ s✐♥❣❧❡t♦♥ s❡ts✱ ✐t ✐s ❝❧❡❛r t❤❛t %}∗

✐s ❛❧s♦ ❛ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡r✳ ◆♦✇ ❝♦♥s✐❞❡r ❛♥② ♠❡♥✉ A ❛♥❞ ❧❡t x _∈ max (A,%∗_{)✳ ▲❡t}

{x1, ..., xn} ❜❡ ❛♥ ❡♥✉♠❡r❛t✐♦♥ ♦❢ A ✇❤❡r❡ x1 = x✳ ❆ s✐♠♣❧❡ ✐♥❞✉❝t✐✈❡ ❛r❣✉♠❡♥t s❤♦✇s

t❤❛t {x} ∼ {x1, ..., xk}❢♦r ❛♥② k ∈ {1, ..., n}✳ ❲❡ ❧❡❛r♥ t❤❛t x∼A✳ ❚❤❛t ✐s✱ ❛♥② ♠❡♥✉ A

✐s ✐♥❞✐✛❡r❡♥t t♦ t❤❡ s✐♥❣❧❡t♦♥{x}✱ ✇❤❡♥❡✈❡r x∈max (A,%∗_{)✳ ◆♦✇ ✜① ❛♥② t✇♦ ♠❡♥✉s A}

❛♥❞B ❛♥❞ ♣✐❝❦xA∈max (A,%∗_{)❛♥❞xB ∈max (B,%}∗_{)✳ ❇② ✇❤❛t ✇❡ ❤❛✈❡ ❥✉st ♦❜s❡r✈❡❞✱}

✷✳ ❘❊❱❊❆▲❊❉ P❘❊❋❊❘❊◆❈❊ ❚❍❊❖❘❨ ✶✵ ❊①❡r❝✐s❡ ✷✳✻ ✭❇❛s❡❞ ♦♥ ❆✐③❡r♠❛♥ ❛♥❞ ▼❛❧✐s❤❡✈s❦✐ ✭✶✾✽✶✮ ❛♥❞ ❋✉rt❛❞♦ ❡t ❛❧✳ ✭✷✵✶✻✮✮✳ ▲❡t

X ❜❡ ❛ ✜♥✐t❡ s❡t ❛♥❞ ❝♦♥s✐❞❡r ❛ ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ c ♦♥ (X,2X _{\ {∅})✳ ❲❡ s❛② t❤❛t c ❤❛s ❛}

♣s❡✉❞♦✲r❛t✐♦♥❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝♦❧❧❡❝t✐♦♥ R♦❢ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡rs ♦♥ X s✉❝❤ t❤❛t✱

❢♦r ❡✈❡r② ♥♦♥❡♠♣t② s✉❜s❡t A ♦❢ X✱

c(A) = [

%∈R

max(A,%).

❙❤♦✇ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦st✉❧❛t❡ ❝❤❛r❛❝t❡r✐③❡s t❤❡ ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡s t❤❛t ❛❞♠✐t s✉❝❤ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ t❤✐s s❡t✉♣✿

❆①✐♦♠ ✺ ✭Ps❡✉❞♦✲❲❆❘P✮✳ ❋♦r ❛❧❧ ♥♦♥❡♠♣t② A, B ⊆X✱ ✐❢ x∈c(B)∩A ❛♥❞ c(B∪ {y})⊆B

❢♦r ❡✈❡r② y ∈A✱ t❤❡♥ x∈c(A)✳

❙♦❧✉t✐♦♥✳ ❙✉♣♣♦s❡ ✜rst t❤❛t c ✐s ❛ ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ t❤❛t ❤❛s ❛ ♣s❡✉❞♦✲r❛t✐♦♥❛❧ r❡♣r❡✲

s❡♥t❛t✐♦♥ R✳ ❋✐① A, B _⊆ X✳ ❙✉♣♣♦s❡ x _∈ c(B)_∩A ❛♥❞ c(B_{∪ {}y_})_⊆ B ❢♦r ❡✈❡r② y _∈ A✳ ❙✐♥❝❡ x∈c(B), ✇❡ ❦♥♦✇ t❤❛t t❤❡r❡ ❡①✐sts%_{∈ R} s✉❝❤ t❤❛tx%y ❢♦r ❡✈❡r②y∈B✳ ■❢✱ ❢♦r ❛♥②y∈A\B✱

✇❡ ❤❛❞ y % x✱ t❤❡♥ ✇❡ ✇♦✉❧❞ ❤❛✈❡ y ∈c(B∪ {y})✱ ✇❤✐❝❤ ✇❡ ❦♥♦✇ ✐s ♥♦t tr✉❡✳ ❇✉t t❤❡♥✱ x ≻ y

❢♦r ❡✈❡r② y ∈ A\B✳ ❙✐♥❝❡ ✇❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t x % y ❢♦r ❡✈❡r② y ∈ B✱ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ c

♥♦✇ ✐♠♣❧✐❡s t❤❛t x∈c(A)✳ ❚❤✐s s❤♦✇s t❤❛t cs❛t✐s✜❡s Ps❡✉❞♦✲❲❆❘P✳

❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡c✐s ❛ ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ t❤❛t s❛t✐s✜❡s Ps❡✉❞♦✲❲❆❘P✳ ❋✐① ❛♥② ♥♦♥❡♠♣t② A _⊆X ❛♥❞ x_∈ c(A)✳ ❖✉r ❣♦❛❧ ✐s t♦ ❝♦♥str✉❝t ❛ ❧✐♥❡❛r ♦r❞❡r <x,A s✉❝❤ t❤❛t {x} = max(A,<x,A)

❛♥❞✱ ❢♦r ❡✈❡r② ♥♦♥❡♠♣t② B _⊆ X✱ max(B,<x,A)_⊆ c(B)✳ ❲❡ ✇✐❧❧ ❞♦ t❤❛t ✐♥❞✉❝t✐✈❡❧②✳ ❙t❛rt ✇✐t❤ X✳ ■❢ x _∈ c(X)✱ ♠❛❦❡ x1 :=x, ♦t❤❡r✇✐s❡✱ ♣✐❝❦ ❛♥② x1 ∈ c(X)\A✳ ◆♦t❡ t❤❛t s✉❝❤ x1 ♠✉st ❡①✐st

❛s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ Ps❡✉❞♦✲❲❆❘P✳ ◆♦✇ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❞✉❝t✐✈❡ ♣r♦❝❡❞✉r❡✳ ❋♦r

k _∈ N, ✐❢ xi ₆= x ❢♦r ❡✈❡r② i _≤ k✱ ♠❛❦❡ xk+1 := x ✇❤❡♥❡✈❡r x ∈ c(X \ {x1, . . . , xk}) ❛♥❞ ♠❛❦❡

xk+1 := y ❢♦r ❛♥② y ∈ c(X \ {x1, . . . , xk})\A ✇❤❡♥ x /∈ c(X\ {x1, . . . , xk})✳ ■❢ xi = x ❢♦r s♦♠❡

i _≤ k✱ s✐♠♣❧② ♣✐❝❦ ❛♥② xk+1 ∈ c(X \ {x1, . . . , xk})✳ ❆❢t❡r t❤❛t✱ ❞❡✜♥❡ <x,A ❛s t❤❡ ❧✐♥❡❛r ♦r❞❡r

t❤❛t s❛②s t❤❛tx1 <x,A x2 <x,A . . .✳ ❆ ❢❡✇ ♦❜s❡r✈❛t✐♦♥s ❛r❡ ✐♥ ♦r❞❡r✳ ❋✐rst✱ ♦❜s❡r✈❡ t❤❛t✱ ❢♦r ❡✈❡r② k ∈ N✱ ✇❡ ❤❛✈❡ t❤❛t xk ∈ c(X \ {x1, . . . , xk−1})✱ ✇✐t❤ t❤❡ ❝♦♥✈❡♥t✐♦♥ t❤❛t {x1, . . . , xk−1} := ∅

✇❤❡♥ k = 1✳ ❇② Ps❡✉❞♦✲❲❆❘P✱ t❤✐s ✐♠♣❧✐❡s t❤❛t xk ∈ c(B) ❢♦r ❡✈❡r② B ⊆ X \ {x1, . . . , xk}✳

❚❤✐s s❤♦✇s t❤❛t✱ ❢♦r ❡✈❡r② ♥♦♥❡♠♣t② B ⊆ X✱ ✇❡ ❤❛✈❡ max(B,<x,A) ⊆ B✳ ❙❡❝♦♥❞✱ s✉♣♣♦s❡ t❤❛t xk = x✱ ❢♦r s♦♠❡ k ∈ N✳ ❇② ❝♦♥str✉❝t✐♦♥✱ ✇❡ ❤❛✈❡ t❤❛t xi ∈/ A ❢♦r ❡✈❡r② i < k✳ ❚❤✐s ✐♠♣❧✐❡s

t❤❛t A ⊆ {xk, xk+1, . . . , x|X|}✱ s♦ t❤❛t max(A,<x,A) = {x}✳ ◆♦✇ ✐t ✐s ❝❧❡❛r t❤❛t ✐❢ ✇❡ ❞❡✜♥❡

R:=_{<x,A:A _∈2X _{\ {∅} ❛♥❞ x∈c(A)}✱ t❤❡♥R ✐s ❛ ♣s❡✉❞♦✲r❛t✐♦♥❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ c✳}

❊①❡r❝✐s❡ ✷✳✼ ✭❇❛s❡❞ ♦♥ ❋✉rt❛❞♦ ❡t ❛❧✳ ✭✷✵✶✻✮✮✳ ▲❡t X ❜❡ ❛ ✜♥✐t❡ s❡t ❛♥❞ ❝♦♥s✐❞❡r ❛ ❝❤♦✐❝❡

❝♦rr❡s♣♦♥❞❡♥❝❡c♦♥(X,2X _{\ {∅})✳ ❲❡ s❛② t❤❛t c❤❛s ❛ ✇❡❛❦ ❝❛t❡❣♦r✐③❛t✐♦♥ r❡♣r❡s❡♥t❛t✐♦♥ ✐❢ t❤❡r❡}

❡①✐sts ❛ ❝♦❧❧❡❝t✐♦♥ S ♦❢ ♥♦♥❡♠♣t② s✉❜s❡ts ♦❢ X ❛♥❞ ❛ ♣r❡♦r❞❡r % ♦♥ X s✉❝❤ t❤❛t % ✐s ❝♦♠♣❧❡t❡

✐♥s✐❞❡ ❡❛❝❤S _{∈ S} ❛♥❞✱ ❢♦r ❡✈❡r② ♥♦♥❡♠♣t② s✉❜s❡t A ♦❢ X✱ c(A) = [

S∈S

max(A∪S,%).

❉❡✜♥❡ t❤❡ ❜✐♥❛r② r❡❧❛t✐♦♥ ⊲_⊆ X _× X ❜② x ⊲ y ✐❢ t❤❡r❡ ❡①✐sts A _⊆ X ✇✐t❤ y _∈ c(A)✱ ❜✉t y /_∈ c(A_{∪ {}x_})✳ ❙❤♦✇ t❤❛t Ps❡✉❞♦✲❲❆❘P ♣❧✉s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦st✉❧❛t❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ❝❤♦✐❝❡

❝♦rr❡s♣♦♥❞❡♥❝❡s t❤❛t ❛❞♠✐t s✉❝❤ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ t❤✐s s❡t✉♣✿

❆①✐♦♠ ✻ ✭❆❝②❝❧✐❝✐t②✮✳ ❚❤❡ r❡❧❛t✐♦♥⊲✐s ❛❝②❧✐❝✐❝✳ ❚❤❛t ✐s✱ ❢♦r ♥♦ ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥_{x1, . . . , xn} ⊆

X✱ ✇❡ ❤❛✈❡ x1 ⊲x2 ⊲· · ·⊲xn⊲x1✳

❙♦❧✉t✐♦♥✳ ❙✉♣♣♦s❡ ✜rst t❤❛t c ✐s ❛ ❝❤♦✐❝❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ t❤❛t ❤❛s ❛ ✇❡❛❦ ❝❛t❡❣♦r✐③❛t✐♦♥

✷✳ ❘❊❱❊❆▲❊❉ P❘❊❋❊❘❊◆❈❊ ❚❍❊❖❘❨ ✶✶ ■❢✱ ❢♦r ❛♥② y ∈ (A\B)∩S✱ ✇❡ ❤❛❞ y % x✱ t❤❡♥ ✇❡ ✇♦✉❧❞ ❤❛✈❡ y ∈ c(B∪ {y})✱ ✇❤✐❝❤ ✇❡ ❦♥♦✇

✐s ♥♦t tr✉❡✳ ❇✉t t❤❡♥✱ x _≻ y ❢♦r ❡✈❡r② y _∈ (A_\B)_∩S✳ ❙✐♥❝❡ ✇❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t x % y ❢♦r

❡✈❡r② y _∈ B _∩S✱ ✇❡ ❣❡t t❤❛t x % y ❢♦r ❡✈❡r② y _∈ A_∩ S ❛♥❞✱ ❜② t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ c✱ ✇❡

❧❡❛r♥ t❤❛t x_∈c(A)✳ ❚❤✐s s❤♦✇s t❤❛t cs❛t✐s✜❡s Ps❡✉❞♦✲❲❆❘P✳ ❋✐♥❛❧❧②✱ s✉♣♣♦s❡ _{x1, . . . , xn} ❛r❡

s✉❝❤ t❤❛t x1 ⊲ x2 ⊲ · · · ⊲ xn✳ ❋r♦♠ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ c✱ ✐t ✐s ❝❧❡❛r t❤❛t t❤✐s ✐♠♣❧✐❡s t❤❛t

x1 ≻x2 ≻ · · · ≻xn✳ ❙✐♥❝❡ %✐s tr❛♥s✐t✐✈❡✱ t❤✐s ♥♦✇ ✐♠♣❧✐❡s t❤❛t x1 ≻xn✳ ❋r♦♠ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥

♦❢ c✐t ✐s ♥♦✇ ❝❧❡❛r t❤❛t x1 ∈c(A∪ {xn}) ❢♦r ❡✈❡r② ❝❤♦✐❝❡ ♣r♦❜❧❡♠ A s✉❝❤ t❤❛t x1 ∈c(A)✳ ❚❤❛t

✐s✱ ✐t ❝❛♥♥♦t ❜❡ t❤❡ ❝❛s❡ t❤❛t xn⊲x1✳

❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ c ✐s ❛ ❝❤♦✐❝❡ ❢✉♥❝t✐♦♥ t❤❛t s❛t✐s✜❡s Ps❡✉❞♦✲❲❆❘P ❛♥❞ ❆❝②❝❧✐❝✐t②✳ ❙✐♥❝❡

t❤❡ r❡❧❛t✐♦♥ ⊲ ✐s ❛❝②❝❧✐❝✱ ✇❡ ❦♥♦✇ t❤❛t tran(⊲) ✐s ❛ str✐❝t ♣❛rt✐❛❧ ♦r❞❡r t❤❛t ❡①t❡♥❞s ⊲✳ ❇② t❤❡

❙③♣✐❧r❛❥♥✬s ❚❤❡♦r❡♠✱ t❤❡r❡ ❡①✐sts ❛ ❧✐♥❡❛r ♦r❞❡r < t❤❛ts ❡①t❡♥❞s tran(⊲) ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ ❛❧s♦

❡①t❡♥❞s ⊲✳ ❋✐① ❛♥② ♥♦♥❡♠♣t② A⊆X ❛♥❞ x∈c(A)✳ ❖✉r ❣♦❛❧ ✐s t♦ ✜♥❞ ❛ s❡t Sx,A ⊆X s✉❝❤ t❤❛t

{x}= max(A∩Sx,A,<)❛♥❞✱ ❢♦r ❡✈❡r② ♥♦♥❡♠♣t②B ⊆X✱max(B∩Sx,A,<)⊆c(B)✳ ❋♦r t❤❛t✱ ❞❡✜♥❡ Sx,A :={x}∪{y∈X\A:y<x ❛♥❞ y ∈c(A∪{y})}✳ ◆♦t✐❝❡ t❤❛tA∩Sx,A ={x}✱ s♦ ✐t ✐s ✐♠♠❡❞✐❛t❡

t❤❛t {x_} = max(A_∩Sx,A,<)✳ ◆♦✇ ✜① ❛♥② ♥♦♥❡♠♣t② B _⊆ X ❛♥❞ ❧❡t _{y_} = max(B _∩Sx,A,<)✳

❲❡ ✇✐❧❧ s❤♦✇ t❤❛t y _∈c(A_∪B)✳ ❚♦ s❡❡ t❤❛t✱ r❡❝❛❧❧ t❤❛t✱ ❜② ❝♦♥str✉❝t✐♦♥✱ y _∈ c(A_{∪ {}y_})✳ ❙♦✱ ✐❢ y /_∈c(A_∪B) ✱ t❤❡r❡ ♠✉st ❡①✐st D_⊆B ❛♥❞ z _∈B_\D ✇✐t❤ y_∈c(A_∪D),❜✉t y /_∈c(A_∪D_{∪ {}z_})✳

❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ⊲✱ t❤✐s ✐♠♣❧✐❡s t❤❛t z ⊲ y ❛♥❞✱ s✐♥❝❡ < ❡①t❡♥❞s ⊲✱ ✇❡ ❣❡t t❤❛t z _≻ y < x✳

▼♦r❡♦✈❡r✱ ❜② Ps❡✉❞♦✲❲❆❘P✱ t❤✐s ❝❛♥ ❤❛♣♣❡♥ ♦♥❧② ✐❢z _∈c(A_∪D_∪{z_})✳ ❆♣♣❧②✐♥❣ Ps❡✉❞♦✲❲❆❘P

♦♥❡ ♠♦r❡ t✐♠❡✱ ✇❡ ❣❡t t❤❛t z _∈ c(A_{∪ {}z_})✳ ❚❤❛t ✐s✱ z _∈ Sx,A✳ ❚❤✐s ❝♦♥tr❛❞✐❝ts t❤❡ ❢❛❝t t❤❛t

{y_} = max(B _∩ Sx,A,<)✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t y _∈ c(A _∪B) ❛♥❞✱ ❜② Ps❡✉❞♦✲❲❆❘P✱

❈❍❆P❚❊❘ ✸

❘❡♣r❡s❡♥t❛t✐♦♥ ❜② ❯t✐❧✐t② ❋✉♥❝t✐♦♥

❊①❡r❝✐s❡ ✸✳✶✳ ❙✉♣♣♦s❡ t❤❛t (X,%)✐s ❛ ♣r❡♦r❞❡r❡❞ s❡t ❛♥❞ % ✐s ❝♦♠♣❧❡t❡✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t

Y ⊆X ✐s ❝♦✉♥t❛❜❧❡ ❛♥❞ %✲❞❡♥s❡ ✐♥ X✳ ❙✐♥❝❡ Y ✐s ❝♦✉♥t❛❜❧❡✱ ❜② ❛ ♣r❡✈✐♦✉s r❡s✉❧t✱ ✇❡ ❦♥♦✇ t❤❛t

t❤❡r❡ ❡①✐sts ❛ ✉t✐❧✐t② ❢✉♥❝t✐♦♥v :Y →R s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ x, y ∈Y✱ x%y ⇐⇒ v(x)≥v(y).

✭❛✮ ❖♥❡ ♠❛② ❝♦♥❥❡❝t✉r❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥ u:X →R ♠✐❣❤t ❜❡ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r %♦♥ X✿

u(x) := sup_{v(y) :y_∈Y ❛♥❞ x%y_}.

●✐✈❡ ❛♥ ❡①❛♠♣❧❡ s❤♦✇✐♥❣ t❤❛t s✉❝❤ ❛ ❢✉♥❝t✐♦♥ u ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② r❡♣r❡s❡♥t %✳

✭❜✮ ■t ✐s ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t② t♦ ❛ss✉♠❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ v ✐s ❜♦✉♥❞❡❞ ❛❜♦✈❡ ❛♥❞ ❜❡❧✲

❧♦✇✳✶,✷_{✳ ■♥ ❢❛❝t✱ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛tv(Y)⊆(0,1)✳ ❲❤❛t ❛❜♦✉t}

t❤❡ ❢✉♥❝t✐♦♥ u:X →R ❞❡✜♥❡❞ ❜②

u(x) := sup{v(y) :y∈Y ❛♥❞ x%y}+ inf{v(y) :y∈Y ❛♥❞ y%x}

2 ,

✇✐t❤ t❤❡ ❝♦♥✈❡♥t✐♦♥ t❤❛tsup (∅) = inf{v(y) : y∈Y} ❛♥❞ inf (∅) = sup{v(y) :y∈Y}❄ ■s

✐t ❛❧✇❛②s ❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ %❄

❙♦❧✉t✐♦♥✳

✭❛✮ ❈♦♥s✐❞❡r t❤❡ s❡t X := [0,1]_∪√

2 ✉♥❞❡r t❤❡ ✉s✉❛❧ ♦r❞❡r ≥✳ ■t ✐s ♥♦t ❤❛r❞ t♦ s❡❡ t❤❛t

t❤❡ s❡tY :=Q_∩[0,1] ✐s≥✲❞❡♥s❡ ✐♥ X ❛♥❞ t❤❛t t❤❡ ❢✉♥❝t✐♦♥v s✉❝❤ t❤❛tv(y) = y ❢♦r ❛❧❧ y ∈ Y r❡♣r❡s❡♥ts t❤❡ r❡str✐❝t✐♦♥ ♦❢ ≥ t♦ Y✳ ❍♦✇❡✈❡r✱ sup

v(y) :y∈Y ❛♥❞ √2≥y = 1 = sup_{v(y) :y_∈Y ❛♥❞ 1_≥y_}✱ ✇❤✐❝❤ s❤♦✇s t❤❛t t❤❡ ❝♦♥str✉❝t✐♦♥ ♣r♦♣♦s❡❞ ✐♥ t❤❡

q✉❡st✐♦♥ ✇♦✉❧❞ ♥♦t ✇♦r❦ ✐♥ ❣❡♥❡r❛❧✳

✭❜✮ ❚❤❡ ❛♥s✇❡r ✐s ♥♦✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡✿ ▲❡t X := [0,1]_∪√

2 _∪(2,3]✳ ◆♦t❡

t❤❛t t❤❡ s❡t Y :=Q_∩([0,1]_∪(2,3]) ✐s ≥✲❞❡♥s❡ ✐♥ X ❛♥❞ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ v s✉❝❤ t❤❛t v(y) = y ❢♦r ❛❧❧ y _∈ Q_∩[0,1] ❛♥❞ v(y) = y ₋1 ❢♦r ❛❧❧ y _∈ Q _∩(2,3] r❡♣r❡s❡♥ts t❤❡

r❡str✐❝t✐♦♥ ♦❢ ≥t♦ Y✳ ❍♦✇❡✈❡r✱ ✐❢ ✇❡ ❞❡✜♥❡ u ❛s ❛❜♦✈❡ ✇❡ ❤❛✈❡u(1) =u √2

= 1✳

❊①❡r❝✐s❡ ✸✳✷✳ ▲❡tX ❜❡ ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❧❡t%_⊆X_×X ❜❡ ❛ ❝♦♥t✐♥✉♦✉s ❝♦♠♣❧❡t❡

♣r❡❢❡r❡♥❝❡✳ ❇② ❉❡❜r❡✉✬s t❤❡♦r❡♠ ✇❡ ❦♥♦✇ t❤❛t%❛❞♠✐ts ❛ r❡♣r❡s❡♥t❛t✐♦♥ ❜② s♦♠❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥

u✳ ■♥ t✉r♥✱ t❤✐s ✐♠♣❧✐❡s t❤❛t X ❤❛s ❛ ❝♦✉♥t❛❜❧❡ %✲❞❡♥s❡ s✉❜s❡t✳ ●✐✈❡ ❛ ❞✐r❡❝t ♣r♦♦❢ ♦❢ t❤✐s ❢❛❝t✳

❚❤❛t ✐s✱ ♣r♦✈❡ ✐t ✇✐t❤♦✉t ✉s✐♥❣ t❤❡ ❢❛❝t t❤❛t% ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ✉t✐❧✐t② ❢✉♥❝t✐♦♥✳

❙♦❧✉t✐♦♥✳ ❙✐♥❝❡X ✐s ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡✱ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡Y _⊆X ✇❤✐❝❤ ✐s ❞❡♥s❡

✐♥X✳ ❲❡ ❛❧s♦ ❦♥♦✇ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ ❝♦❧❧❡❝t✐♦♥O ♦❢ ♦♣❡♥ s✉❜s❡ts ♦❢X s✉❝❤ t❤❛t ❛♥②

♦♣❡♥ s✉❜s❡t V ♦❢ X ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s V =∪{o ∈ O :o ⊆V}✳ ◆♦✇ ✜① ❛♥② x, z ∈X ✇✐t❤ x≻z✳

■❢ t❤❡r❡ ❡①✐sts w ∈ X ✇✐t❤ x ≻ w ≻ z✱ t❤❡♥ w ∈ L≻(x)∩U≻(z)✱ s♦ t❤❛t L≻(x)∩U≻(z) ✐s ❛

♥♦♥❡♠♣t② ♦♣❡♥ s❡t✱ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ %✳ ❙✐♥❝❡ Y ✐s ❞❡♥s❡ ✐♥ X✱ t❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ❡①✐sts ✶_{❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥v ✐♥ t❤❡ ♠❛✐♥ t❡①t ✐♠♣❧✐❡s t❤❛tv(y)∈(0,1)❢♦r ❛♥②y∈Y✳} ✷_{❚❤❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ v ✐s ♥❡❝❡ss❛r② ♦♥❧② t♦ ❣✉❛r❛♥t❡❡ t❤❛t sup}′s ❛♥❞ _inf′_{s ❛r❡ ♥❡✈❡r ∞ ♥♦r −∞✳ ❇✉t t❤❡}

✸✳ ❘❊P❘❊❙❊◆❚❆❚■❖◆ ❇❨ ❯❚■▲■❚❨ ❋❯◆❈❚■❖◆ ✶✸

y ∈Y ✇✐t❤ x ≻y ≻z✳ ❙♦ ✇❡ ♦♥❧② ♥❡❡❞ t♦ ✇♦rr② ✇✐t❤ t❤❡ ♣❛✐rsx ❛♥❞ z s✉❝❤ t❤❛t x≻ z ❛♥❞ ❢♦r

♥♦ w_∈X ✐t ✐s tr✉❡ t❤❛t x_≻w_≻z✳ ❲❡ ✇✐❧❧ ♥♦✇ s❤♦✇ t❤❛t✱ ✉♣ t♦ ✐♥❞✐✛❡r❡♥❝❡s✱ t❤✐s ❝❛♥ ❤❛♣♣❡♥

♦♥❧② ❛ ❝♦✉♥t❛❜❧❡ ♥✉♠❜❡r ♦❢ t✐♠❡s✳ ▲❡t✬s ✜rst ❜r✐♥❣ t❤✐♥❣s t♦ t❤❡ q✉♦t✐❡♥t s♣❛❝❡✳ ❚❤❛t ✐s✱ ❢♦r ❡❛❝❤

x _∈ X ❞❡✜♥❡ [x]_∼ := _{w _∈ X : x _∼ w_} ❛♥❞ ❞❡✜♥❡ X_{\ ∼}:= _{[x]_∼ : x _∈ X}✳ ❲❡ ✉s❡ t❤❡ s②♠❜♦❧ ˆ

% t♦ r❡♣r❡s❡♥t t❤❡ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦♥X\ ∼ s✉❝❤ t❤❛t [x]_∼%ˆ[y]_∼ ✐✛ x%y ✭❝♦♥✈✐♥❝❡ ②♦✉rs❡❧❢ t❤❛t

❡✈❡r②t❤✐♥❣ ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❤❡r❡✮✳ ◆♦✇✱ ❢♦r ❡❛❝❤ [x]_∼ ∈ X\ ∼ s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts [z]_∼ ∈ X\ ∼ ✇✐t❤ [x]_∼≻ˆ[w]_∼≻ˆ[z]_∼ ❢♦r ♥♦ [w]_{∼ ∈} X_{\ ∼}✱ ♣✐❝❦ s♦♠❡ x _∈ [x]_∼ ❛♥❞ ❧❡t Y˜ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ t❤❡s❡ x✬s✳ ❲❡ ♥♦✇ s❤♦✇ t❤❛tY˜ ✐s ❝♦✉♥t❛❜❧❡✳ ◆♦t❡ t❤❛t✱ ❢♦r ❡❛❝❤x_∈Y˜ ✇❡ ❤❛✈❡x_≻zx ❢♦r s♦♠❡ zx _∈X

s✉❝❤ t❤❛t x _≻ w _≻ zx ❢♦r ♥♦ w _∈ X✳ ❋♦r ❡❛❝❤ s✉❝❤ x ❧❡t ox _{∈ O} ❜❡ s✉❝❤ t❤❛t ox _⊆ L≻(x) ❛♥❞

zx _∈ox✳ ◆♦✇ ♣✐❝❦ ❛♥② t✇♦ ❞✐st✐♥❝t ❛❧t❡r♥❛t✐✈❡sx,x˜_∈Y˜✳ ❙✉♣♣♦s❡✱ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ t❤❛t x_≻x˜✳✸ ❲❡ ♥♦t❡ t❤❛t ox ❛♥❞ ox˜ ♠✉st ❜❡ ❞✐st✐♥❝t s❡ts✱ s✐♥❝❡ zx ∈ox\ox˜✳ ❚❤❡r❡ ✐s✱ t❤❡r❡ ❡①✐sts ❛♥

✐♥❥❡❝t✐✈❡ ♠❛♣ ❢r♦♠ _Y˜ _{✐♥t♦ O✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t Y}˜ _{✐s ❝♦✉♥t❛❜❧❡✳ ❇✉t✱ ❢♦r ❡❛❝❤x, z ∈X s✉❝❤ t❤❛t} x_≻z ❛♥❞ ❢♦r ♥♦ w _∈X ✐t ✐s tr✉❡ t❤❛t x_≻w_≻ z t❤❡r❡ ❡①✐stsy _∈[x]_∼∩Y˜✳ ❚❤❛t ✐s✱ t❤❡r❡ ❡①✐sts y_∈Y˜ ✇✐t❤ x_∼y✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❝♦✉♥t❛❜❧❡ s❡t Y _∪Y˜ ✐s %✲❞❡♥s❡ ✐♥ X.

❊①❡r❝✐s❡ ✸✳✸✳ ▲❡t X ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❧❡t v : X _→ R ❜❡ ❛♥② ❢✉♥❝t✐♦♥✳ ❚❤❡ lim sup ♦❢

t❤❡ ❢✉♥❝t✐♦♥ v ✐s ❞❡✜♥❡❞ ❜②

v•(x) := lim

m→∞sup{v(y) :y∈BX(x,

1 m)}.

✭❛✮ ❙❤♦✇ t❤❛t v• _{✐s ❛❧✇❛②s ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s ❢♦r ❛♥② ❢✉♥❝t✐♦♥ v✳}

✭❜✮ ❘❡❝❛❧❧ t❤❡ ❢✉♥❝t✐♦♥ u˜ t❤❛t ✇❡ ❝r❡❛t❡❞ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ❘❛❞❡r✬s ❯t✐❧✐t② ❘❡♣r❡s❡♥t❛t✐♦♥

❚❤❡♦r❡♠✳ ●✐✈❡ ❛♥ ❡①❛♠♣❧❡ s❤♦✇✐♥❣ t❤❛t u˜ ✐s ♥♦t ♥❡❝❡ss❛r✐❧② ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s✳

✭❝✮ ❖❜✈✐♦✉s❧②✱ t❤❡ lim sup✱ u˜•_{✱ ♦❢ u˜ ✐s ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s✳ ●✐✈❡ ❛♥ ❡①❛♠♣❧❡ s❤♦✇✐♥❣ t❤❛t✱}

❤♦✇❡✈❡r✱ ✐t ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② r❡♣r❡s❡♥t t❤❡ ❝♦♠♣❧❡t❡ ♣r❡♦r❞❡r % ✐♥ ❘❛❞❡r✬s ❯t✐❧✐t②

❘❡♣r❡s❡♥t❛t✐♦♥ ❚❤❡♦r❡♠✳ ❙♦❧✉t✐♦♥✳

✭❛✮ ▲❡tX ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡✱ ✜① ❛♥② ❢✉♥❝t✐♦♥v :X →R❛♥❞ ❞❡✜♥❡ v• _{❛s ❛❜♦✈❡✳ ◆♦✇✱ ✜① ❛♥②}

ε > 0 ❛♥❞ x _∈ X✳ ❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ v•_{✱ t❤❡r❡ ❡①✐sts m ∈ N s✉❝❤ t❤❛t sup{v(y) : y ∈}

BX(x, 1

m)}< v

•_{(x) +ε✳ ◆♦t❡ t❤❛t✱ ❜② t❤❡ tr✐❛♥❣✉❧❛r ✐♥❡q✉❛❧✐t②✱ ✇❡ ❤❛✈❡ t❤❛tBX(z,} 1 2m)⊆ BX(x,_m1)❢♦r ❡✈❡r② z ∈BX(x,₂1_m)✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t

v•(z) _≤ sup_{v(y) :y_∈BX(z, 1 2m)}

≤ sup{v(y) :y∈BX(x, 1 m)} < v•(x) +ε,

❢♦r ❡✈❡r② z _∈BX(x, 1

2m) ✳ ❚❤✐s s❤♦✇s t❤❛t v

• _{✐s ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s ❛tx✳}

✭❜✮ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t X ✐s ❛ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❧❡t _O :=_{o1, o2, . . .} ❜❡ ❛ ❜❛s✐s ❢♦r

X✬s t♦♣♦❧♦❣②✳ ❘❡❝❛❧❧ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ u˜ ✇❛s ❞❡✜♥❡❞ ❜② ˜

u(x) := X

{i:oi⊆L≻(x)}

1 2i,

❢♦r ❡✈❡r②x _∈X✳ ◆♦✇ ❧❡t X := [0,1]_{∪ {}2_} ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ♣r❡♦r❞❡r % s✉❝❤ t❤❛t x% y

✐✛ x _≥ y✱ ✐❢ x, y _∈ [0,1] ❛♥❞ 0 _∼ 2✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t % ✐s ❯❙❈✳ ❙✉♣♣♦s❡ t❤❛t

O =_{o1, o2, . . .} ✐s ❛ ❝♦✉♥t❛❜❧❡ ❜❛s✐s ❢♦r X✬s t♦♣♦❧♦❣②✳ ❲❡ ♥♦t❡ t❤❛t ✐t ♠✉st ❜❡ t❤❡ ❝❛s❡

✸✳ ❘❊P❘❊❙❊◆❚❆❚■❖◆ ❇❨ ❯❚■▲■❚❨ ❋❯◆❈❚■❖◆ ✶✹ t❤❛t {2} ∈ O✳ ❚❤❛t ✐s✱ oi∗ = {2} ❢♦r s♦♠❡ i∗ ∈ N✳ ❚❤✐s ♥♦✇ ✐♠♣❧✐❡s t❤❛t✱ ❢♦r ❡✈❡r②

x_∈(0,1]✱u(x)˜ _≥u(0) +˜ 1

2i∗ ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱u˜ ✐s ♥♦t ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s ❛t 0✳

✭❝✮ ❲❡ ❤❛✈❡ s❡❡♥ ❛❜♦✈❡ t❤❛t✱ ❢♦r ❡✈❡r② x _∈ (0,1]✱ u(x)˜ _≥ u(0) +˜ 1

2i∗✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t

˜

u•_(0)≥ 1

2i∗✳✹ ❙✐♥❝❡ ✐t ✐s ❝❧❡❛r t❤❛tu˜•(2) = 0✱ t❤✐s s❤♦✇s t❤❛tu˜• ❞♦❡s ♥♦t r❡♣r❡s❡♥t%✳

❊①❡r❝✐s❡ ✸✳✹ ✭▼✉❧t✐♣❧❡ ▼✉❧t✐♣❧❡ ❯t✐❧✐t✐❡s✱ ✐♥s♣✐r❡❞ ❜② ▲❡❤r❡r ❛♥❞ ❚❡♣❡r ✭✷✵✶✶✮ ❛♥❞ ◆✐s❤✐♠✉r❛ ❛♥❞ ❖❦ ✭✷✵✶✺✮✮✳ ▲❡t X ❜❡ ❛ ❣❡♥❡r✐❝ s❡t ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t②♣❡ ♦❢ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r ❛

❜✐♥❛r② r❡❧❛t✐♦♥ %_⊆X _×X✿ ❚❤❡r❡ ❡①✐sts ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ s❡ts ♦❢ ❢✉♥❝t✐♦♥s _U✱ ✇✐t❤ ❡❛❝❤ ❡❧❡♠❡♥t U

♦❢ U ❜❡✐♥❣ ❛ s✉❜s❡t ♦❢ RX _{s✉❝❤ t❤❛t✱ ❢♦r ❛♥②x, y ∈X✱}

x%y ⇐⇒ ∃U ∈ U ✇✐t❤ u(x)≥u(y) ❢♦r ❛❧❧ u∈U.

❈❤❛r❛❝t❡r✐③❡ t❤❡ ❜✐♥❛r② r❡❧❛t✐♦♥s t❤❛t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛❜♦✈❡✳ ✭❍✐♥t✿ ■t ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ t✇♦ ❧✐♥❡s✮✳

❙♦❧✉t✐♦♥✳ ❆❝t✉❛❧❧②✱ t❤❡ ♦♥❧② r❡str✐❝t✐♦♥ t❤❛t t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❛❜♦✈❡ ✐♠♣♦s❡s ♦♥ ❛ ❜✐♥❛r② r❡❧❛t✐♦♥ ✐s t❤❛t ✐t ❜❡ r❡✢❡①✐✈❡✳ ❚❤❛t ✐s✱ ❛ ❜✐♥❛r② r❡❧❛t✐♦♥ % ❤❛s ❛ r❡♣r❡s❡♥t❛t✐♦♥ ❛s ❛❜♦✈❡ ✐❢ ❛♥❞

♦♥❧② ✐❢ ✐t ✐s r❡✢❡①✐✈❡✳ ❚♦ s❡❡ t❤❛t✱ ✜rst ♥♦t❡ t❤❛t ✐t ✐s ❝❧❡❛r t❤❛t ❛ ❜✐♥❛r② r❡❧❛t✐♦♥ r❡♣r❡s❡♥t❡❞ ❛s ❛❜♦✈❡ ✐s r❡✢❡①✐✈❡✳ ❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ t❤❛t % ✐s ❛ r❡✢❡①✐✈❡ ❜✐♥❛r② r❡❧❛t✐♦♥✳ ❋♦r ❡❛❝❤ (x, y) _∈%✱

❞❡✜♥❡ t❤❡ s❡t ♦❢ ❢✉♥❝t✐♦♥s U(x,y) ⊆RX ❜②

U(x,y):=

u∈RX :u(x)≥u(y) .

■t ✐s ❝❧❡❛r t❤❛t t❤❡ ❝♦❧❧❡❝t✐♦♥ U :=_{U(x,y): (x, y)∈%} r❡♣r❡s❡♥ts% ✐♥ t❤❡ s❡♥s❡ ❛❜♦✈❡✳

❊①❡r❝✐s❡ ✸✳✺ ✭■♥s♣✐r❡❞ ❜② ◆✐s❤✐♠✉r❛ ❛♥❞ ❖❦ ✭✷✵✶✺✮✮✳ ❙❤♦✇ t❤❛t ✐❢ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ s♣❛❝❡X

✐♥ ❊①❡r❝✐s❡ ✸✳✹ ✐s ❛ ♠❡tr✐❝ s♣❛❝❡✱ t❤❡♥ t❤❡ s❛♠❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❞❡❧✐✈❡rs t❤❡ s❛♠❡ r❡♣r❡s❡♥t❛t✐♦♥✱ ❜✉t ✇✐t❤ ❛❧❧ t❤❡ ❢✉♥❝t✐♦♥s t❤❛t ❛♣♣❡❛r ✐♥ t❤❡ ▼✉❧t✐♣❧❡ ▼✉❧t✐♣❧❡ ❯t✐❧✐t✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ❜❡✐♥❣ ❝♦♥t✐♥✉♦✉s✳ ❚❤❛t ✐s✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t U ✐♥ t❤❛t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ✐s ❛ ♥♦♥❡♠♣t② s✉❜s❡t ♦❢

2C(X)_{\ {∅}✳ ✭❍✐♥t✿ ❯s❡ t❤❡ ❯r②s♦❤♥✬s ▲❡♠♠❛ ✭s❡❡ t❤❡ ❯s❡❢✉❧ ▼❛t❤❡♠❛t✐❝❛❧ ❘❡s✉❧ts ❛♣♣❡♥❞✐①✮✮✳}

✹_{■♥ ❢❛❝t✱ ✇❡ ❤❛✈❡ t❤❛tu˜}•₍₀₎✐s ❡①❛❝t❧② ❡q✉❛❧ t♦ 1