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r t str qt tr r rs rst
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐
❯♥✐✈❡rs✐t② ♦❢ ❙✐❧❡s✐❛ ✐♥ ❑❛t♦✇✐❝❡ ❥♦✐♥t ♣❛♣❡r ✇✐t❤ ❏✉❞②t❛ ❇→❦ ✭❛r❳✐✈✿✶✽✵✻✳✵✵✼✽✺✮
◆♦✈✐ ❙❛❞ ✷✵✶✼
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❆ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ s❡t (P, ⊑) ✐s ❛ ❞❝♣♦ ✭❞✐r❡❝t❡❞ ❝♦♠♣❧❡t❡✮✱ ✐❢ ❡✈❡r② ❞✐r❡❝t❡❞ s✉❜s❡t D ⊆ P ❤❛s ❛ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞✱ ❞❡♥♦t❡❞ ❜② D✳ ■♥ ❛ ♣♦s❡t (P, ⊑) a ≪ b ✭ ✏❛ ✐s ❛♣♣r♦①✐♠❛t❡s ❜✑✮ ✐❢ ❢♦r ❡❛❝❤ ❞✐r❡❝t❡❞ s❡t D ⊆ P b ⊑
❆ ❞❝♣♦ P ✐s s❛✐❞ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s ✐❢ ↓ ↓(a) = {b ∈ P : b ≪ a} ✐s ❞✐r❡❝t❡❞ ❛♥❞ ❤❛s a = (↓ ↓(a)) ❢♦r ❡❛❝❤ a ∈ P ✳ ❆ ❞♦♠❛✐♥ ✐s ❝♦♥t✐♥✉♦✉s ❞❝♣♦✳ ❆ s✉❜s❡t U ♦❢ ❛ ♣♦s❡t P ✐s ❙❝♦tt✲♦♣❡♥ ✐❢ U ✐s ❛♥ ✉♣♣❡r s❡t✿ x ∈ U ❛♥❞ x ⊑ y t❤❡♥ y ∈ U✱ ❢♦r ❡✈❡r② ❞✐r❡❝t❡❞ D ⊆ P ✇❤✐❝❤ ❤❛s ❛ s✉♣r❡♠✉♠✱
❉♦♠❛✐♥s ✇❡r❡ ❞✐s❝♦✈❡r❡❞ ✐♥ ❝♦♠♣✉t❡r s❝✐❡♥❝❡ ❜② ❉✳ ❙❝♦tt ✐♥ ✶✾✼✵✳
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❲❤❡♥ ❛ s♣❛❝❡ X ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ t❤❡ s♣❛❝❡ max(P) ❢♦r ❛ ❞♦♠❛✐♥ (P, ⊑) ✇✐t❤ ❙❝♦tt t♦♣♦❧♦❣② ✐♥❤❡r✐t❡❞ ❢r♦♠ P✱ ▼❛rt✐♥ ✇r✐t❡s t❤❛t X ❤❛s ❛ ♠♦❞❡❧✱ ✇❤✐❧❡ ❇❡♥♥❡tt ❛♥❞ ▲✉t③❡r ✇r✐t❡ t❤❛t X ✐s ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡✳ ❑✳ ▼❛rt✐♥✱ ✑❚♦♣♦❧♦❣✐❝❛❧ ❣❛♠❡s ✐♥ ❞♦♠❛✐♥ t❤❡♦r②✑✱ ✷✵✵✸ ❍✳ ❇❡♥❡tt✱ ❉✳ ▲✉t③❡r✱ ✑❙tr♦♥❣ ❝♦♠♣❧❡t♥❡ss Pr♦♣❡rt✐❡s ✐♥ ❚♦♣♦❧♦❣②✑✱ ✷✵✵✾
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
R ✐s ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ P = {[a, b] : a b} [a, b] ⊑ [c, d] ⇔ [c, d] ⊆ [a, b] [a, b] ≪ [c, d] ⇔ [c, d] ⊆ (a, b)
❢♦r ❛♥② ❞✐r❡❝t❡❞ s❡t D ⊆ P max P = {[x, x] : x ∈ R} ❛♥❞ h: max P → R✿ h([x, x]) = x
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❆ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ X ✐s ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ P = {K ⊆ X : ∅ = K ✐s ❝♦♠♣❛❝t} K✶ ⊑ K✷ ⇔ K✷ ⊆ K✶ K✶ ≪ K✷ ⇔ K✷ ⊆ ✐♥tK✶
❢♦r ❛♥② ❞✐r❡❝t❡❞ s❡t D ⊆ P max P = {{x} : x ∈ X} ❛♥❞ h: max P → X h({x}) = x
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❲✳ ❋❧❡✐ss♥❡r✱ ▲✳ ❨❡♥❣✉❧❛❧♣✱ ✑❲❤❡♥ Cp(X) ✐s ❉♦♠❛✐♥ ❘❡♣r❡s❡♥t❛❜❧❡✑✱ ✷✵✶✸
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❲❡ s❛② t❤❛t ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❋✲❨ ✭❋❧❡✐ss♥❡r✕❨❡♥❣✉❧❛❧♣✮ ❝♦✉♥t❛❜❧② ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ ✐❢ t❤❡r❡ ✐s ❛ tr✐♣❧❡ (Q, ≪, B) s✉❝❤ t❤❛t ✭❉✶✮ B : Q → τ ∗(X) ❛♥❞ {B(q) : q ∈ Q} ✐s ❛ ❜❛s❡ ❢♦r τ(X)✱ ✭❉✷✮ ≪ ✐s ❛ tr❛♥s✐t✐✈❡ r❡❧❛t✐♦♥ ♦♥ Q✱ ✭❉✸✮ ❢♦r ❛❧❧ p, q ∈ Q✱ p ≪ q ✐♠♣❧✐❡s B(p) ⊇ B(q)✱ ✭❉✹✮ ❋♦r ❛❧❧ x ∈ X✱ ❛ s❡t {q ∈ Q : x ∈ B(q)} ✐s ❞✐r❡❝t❡❞ ❜② ≪✱ ✭❉✺ω✶✮ ✐❢ D ⊆ Q ❛♥❞ (D, ≪) ✐s ❝♦✉♥t❛❜❧❡ ❛♥❞ ❞✐r❡❝t❡❞✱ t❤❡♥
{B(q) : q ∈ D} = ∅✳
■❢ t❤❡ ❝♦♥❞✐t✐♦♥s ✭❉✶✮✕✭❉✹✮ ❛♥❞ ❛ ❝♦♥❞✐t✐♦♥ ✭❉✺✮ ✐❢ D ⊆ Q ❛♥❞ (D, ≪) ✐s ❞✐r❡❝t❡❞✱ t❤❡♥ {B(q) : q ∈ D} = ∅ ❛r❡ s❛t✐s✜❡❞✱ ✇❡ s❛② t❤❛t ❛ s♣❛❝❡ X ✐s ❋✲❨ ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡✳
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❚❤❡♦r❡♠❬▼❛rt✐♥✱ ✷✵✵✸❪ ❆ ♠❡tr✐❝ s♣❛❝❡ ✐s ❛ ❞♦♠❛✐♥ r❡r❡s❡♥t❛❜❧❡ ✐✛ ✐t ✐s ❝♦♠♣❧❡t❡❧② ♠❡tr✐③❛❜❧❡✳ ❚❤❡♦r❡♠❬❇❡♥❡tt✱ ▲✉t③❡r✱ ✷✵✵✻❪ ■❢ ❛ s♣❛❝❡ ✐s ❷❡❝❤ ❝♦♠♣❧❡t❡✱ t❤❡♥ ✐t ✐s ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡✳ ❚❤❡♦r❡♠❬❇❡♥❡tt✱ ▲✉t③❡r✱ ✷✵✵✻❪ ■❢ ❛ s♣❛❝❡ X ✐s ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ ❛♥❞ ❛ s♣❛❝❡ Y ✐s ❛ Gδ✕s✉❜s♣❛❝❡ ♦❢ X✱ t❤❡♥ Y ✐s ❛ ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ s♣❛❝❡✳
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❲❡ s❛② t❤❛t ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❋✲❨ ✭❋❧❡✐ss♥❡r✕❨❡♥❣✉❧❛❧♣✮ ❝♦✉♥t❛❜❧② π✲❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ ✐❢ t❤❡r❡ ✐s ❛ tr✐♣❧❡ (Q, ≪, B) s✉❝❤ t❤❛t ✭π❉✶✮ B : Q → τ ∗(X) ❛♥❞ {B(q) : q ∈ Q} ✐s ❛ π✲❜❛s❡ ❢♦r τ(X)✱ ✭π❉✷✮ ≪ ✐s ❛ tr❛♥s✐t✐✈❡ r❡❧❛t✐♦♥ ♦♥ Q✱ ✭π❉✸✮ ❢♦r ❛❧❧ p, q ∈ Q✱ p ≪ q ✐♠♣❧✐❡s B(p) ⊇ B(q)✱ ✭π❉✹✮ ✐❢ q, p ∈ Q s❛t✐s❢② B(q) ∩ B(p) = ∅✱ t❤❡r❡ ❡①✐sts r ∈ Q s❛t✐s❢②✐♥❣ p, q ≪ r✱ ✭π❉✺ω✶✮ ✐❢ D ⊆ Q ❛♥❞ (D, ≪) ✐s ❝♦✉♥t❛❜❧❡ ❛♥❞ ❞✐r❡❝t❡❞✱ t❤❡♥
{B(q) : q ∈ D} = ∅✳
■❢ t❤❡ ❝♦♥❞✐t✐♦♥s ✭π❉✶✮✕✭π❉✹✮ ❛♥❞ ❛ ❝♦♥❞✐t✐♦♥ ✭π❉✺✮ ✐❢ D ⊆ Q ❛♥❞ (D, ≪) ✐s ❞✐r❡❝t❡❞✱ t❤❡♥ {B(q) : q ∈ D} = ∅ ❛r❡ s❛t✐s✜❡❞✱ ✇❡ s❛② t❤❛t ❛ s♣❛❝❡ X ✐s ❋✲❨ π✲❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡✳
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❲❡ s❛② t❤❛t ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❋✲❨ ✭❋❧❡✐ss♥❡r✕❨❡♥❣✉❧❛❧♣✮ ❝♦✉♥t❛❜❧② π✲❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ ✐❢ t❤❡r❡ ✐s ❛ tr✐♣❧❡ (Q, ≪, B) s✉❝❤ t❤❛t ✭π❉✶✮ B : Q → τ ∗(X) ❛♥❞ {B(q) : q ∈ Q} ✐s ❛ π✲❜❛s❡ ❢♦r τ(X)✱ ✭π❉✷✮ ≪ ✐s ❛ tr❛♥s✐t✐✈❡ r❡❧❛t✐♦♥ ♦♥ Q✱ ✭π❉✸✮ ❢♦r ❛❧❧ p, q ∈ Q✱ p ≪ q ✐♠♣❧✐❡s B(p) ⊇ B(q)✱ ✭π❉✹✮ ✐❢ q, p ∈ Q s❛t✐s❢② B(q) ∩ B(p) = ∅✱ t❤❡r❡ ❡①✐sts r ∈ Q s❛t✐s❢②✐♥❣ p, q ≪ r✱ ✭π❉✺ω✶✮ ✐❢ D ⊆ Q ❛♥❞ (D, ≪) ✐s ❝♦✉♥t❛❜❧❡ ❛♥❞ ❞✐r❡❝t❡❞✱ t❤❡♥
{B(q) : q ∈ D} = ∅✳
■❢ t❤❡ ❝♦♥❞✐t✐♦♥s ✭π❉✶✮✕✭π❉✹✮ ❛♥❞ ❛ ❝♦♥❞✐t✐♦♥ ✭π❉✺✮ ✐❢ D ⊆ Q ❛♥❞ (D, ≪) ✐s ❞✐r❡❝t❡❞✱ t❤❡♥ {B(q) : q ∈ D} = ∅ ❛r❡ s❛t✐s✜❡❞✱ ✇❡ s❛② t❤❛t ❛ s♣❛❝❡ X ✐s ❋✲❨ π✲❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡✳
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❚❤❡r❡ ❡①✐sts ❛ s♣❛❝❡✱ ✇❤✐❝❤ ✐t ✐s ❋✲❨ ❝♦✉♥t❛❜❧② ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ ✭❋✲❨ ❝♦✉♥t❛❜❧② π✲❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡✮ ❜✉t ✐t ✐s ♥♦t ❋✲❨ π✲❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ ✭ ♥♦t ❋✲❨ ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡✳✮ ❲❡ ❝♦♥s✐❞❡r ❛ s♣❛❝❡ X = Σ
{✵, ✶}ω✶ = x ∈ {✵, ✶}ω✶ : | supp x| ω ,
✇❤❡r❡ supp x = {α ∈ A : x(α) = ✶} ❢♦r x ∈ {✵, ✶}A✱ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✭ω✶✲❜♦① t♦♣♦❧♦❣②✮ ❣❡♥❡r❛t❡❞ ❜② ❛ ❜❛s❡ B =
pr−✶
A (x) : A ∈ [ω✶]ω, x ∈ {✵, ✶}A,
✇❤❡r❡ prA : Σ({✵, ✶}ω✶) → {✵, ✶}A ✐s ❛ ♣r♦❥❡❝t✐♦♥✳
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
Q = B =
pr−✶
A (x) : A ∈ [ω✶]ω, x ∈ {✵, ✶}A,
B : Q → Q ❜❡ t❤❡ ✐❞❡♥t✐t②✳ ❆ r❡❧❛t✐♦♥ ≪ ✐s ❞❡✜♥❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② pr−✶
A (x) ≪ pr−✶ B (y) ⇔ pr−✶ A (x) ⊇ pr−✶ B (y),
❢♦r ❛♥② pr−✶
A (x), pr−✶ B (y) ∈ B✳
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❚✇♦ ♣❧❛②❡rs α ❛♥❞ β ❛❧t❡r♥❛t❡❧② ❝❤♦♦s❡ ♦♣❡♥ ♥♦♥❡♠♣t② s❡ts ✇✐t❤ β U✵ U✶ · · · α V✵ V✶ P❧❛②❡r α ✇✐♥s t❤✐s ♣❧❛② ✐❢ ∞
n=✶ Vn = ∅✳ ❖t❤❡r✇✐s❡ β ✇✐♥s✳ ❉❡♥♦t❡❞
t❤✐s ❣❛♠❡ ❜② BM(X).
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❚✇♦ ♣❧❛②❡rs α ❛♥❞ β ❛❧t❡r♥❛t❡❧② ❝❤♦♦s❡ β U✵ ∋ x✵ U✶ ∋ x✶ · · · α V✵ V✶ P❧❛②❡r α ✇✐♥s ✐❢ {Vn : n ∈ ω} = ∅✳ ❖t❤❡r✇✐s❡ β ✇✐♥s✳
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❆ str❛t❡❣② ❢♦r t❤❡ ♣❧❛②❡r α ✐♥ t❤❡ ❣❛♠❡ BM(X) ♦r Ch(X) ✐s ❛ r✉❧❡ ❢♦r ❝❤♦♦s✐♥❣ ✇❤❛t t♦ ♣❧❛② ♦♥ ❡❛❝❤ r♦✉♥❞ ❣✐✈❡♥ t❤❡ ❢✉❧❧ ✐♥❢♦r♠❛t✐♦♥ ♦❢ ♠♦✈❡s ✉♣ ✉♥t✐❧ t❤❛t ♣♦✐♥t✳ ❆ ✇✐♥♥✐♥❣ str❛t❡❣② ❢♦r t❤❡ ♣❛❧②❡r α ✐s ❛ str❛t❡❣② t❤❛t ♣r♦❞✉❝❡s ❛ ✇✐♥ ❢♦r t❤❛t ♣❧❛②❡r α ✐♥ ❛♥② ❣❛♠❡ ✇❤❡♥ ♣❧❛②✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ t❤❛t st❛rt❡❣②✳
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❚❤❡♦r❡♠❬▼❛rt✐♥✱ ✷✵✵✸❪ ■❢ ❛ s♣❛❝❡ X ✐s ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡✱ t❤❡♥ t❤❡ ♣❧❛②❡r α ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✐♥ Ch(X)✳ ❚❤❡♦r❡♠❬❋❧❡✐ss♥❡r✱ ❨❡♥❣✉❧❛❧♣✱ ✷✵✶✺❪ ■❢ ❛ s♣❛❝❡ X ✐s ❋✲❨ ❝♦✉♥t❛❜❧② ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡✱ t❤❡♥ t❤❡ ♣❧❛②❡r α ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✐♥ Ch(X)✳
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❚❤❡♦r❡♠❬❏✳❇→❦✱ ❆✳ ❑✳❪ ■❢ t❤❡ ♣❧❛②❡r α ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✐♥ Ch(X)✱ t❤❡♥ X ✐s ❋✲❨ ❝♦✉♥t❛❜❧② ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡✳ ❚❤❡♦r❡♠❬❏✳❇→❦✱ ❆✳ ❑✳❪ ❚❤❡ ♣❧❛②❡r α ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✐♥ t❤❡ ❇▼✭X✮ ✐✛ X ✐s ❋✲❨ ❝♦✉♥t❛❜❧② π✲ ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡✳
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②
❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r②