r t str - PowerPoint PPT Presentation

❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r② ❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐ ❯♥✐✈❡rs✐t② ♦❢ ❙✐❧❡s✐❛ ✐♥ ❑❛t♦✇✐❝❡ ❥♦✐♥t ♣❛♣❡r ✇✐t❤ ❏✉❞②t❛ ❇→❦ ✭❛r❳✐✈✿✶✽✵✻✳✵✵✼✽✺✮ ◆♦✈✐ ❙❛❞ ✷✵✶✼ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r② ❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐

❆ ♣❛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 ⊑ D ⇒ ∃ ( d ∈ D ) a ⊑ d . ❆ ❞❝♣♦ 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❡♠✉♠✱ � D ∈ U ⇒ D ∩ U � = ∅ ❉♦♠❛✐♥s ✇❡r❡ ❞✐s❝♦✈❡r❡❞ ✐♥ ❝♦♠♣✉t❡r s❝✐❡♥❝❡ ❜② ❉✳ ❙❝♦tt ✐♥ ✶✾✼✵✳ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r② ❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐

❉♦♠❛✐♥ 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✑✱ ✷✵✵✾ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r② ❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐

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 ) � � D = D ❢♦r ❛♥② ❞✐r❡❝t❡❞ s❡t D ⊆ P max P = { [ x , x ] : x ∈ R } ❛♥❞ h : max P → R ✿ h ([ x , x ]) = x ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r② ❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐

❆ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ X ✐s ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ P = { K ⊆ X : ∅ � = K ✐s ❝♦♠♣❛❝t } K ✶ ⊑ K ✷ ⇔ K ✷ ⊆ K ✶ K ✶ ≪ K ✷ ⇔ K ✷ ⊆ ✐♥t K ✶ � � D = D ❢♦r ❛♥② ❞✐r❡❝t❡❞ s❡t D ⊆ P max P = {{ x } : x ∈ X } ❛♥❞ h : max P → X h ( { x } ) = x ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r② ❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐

❲✳ ❋❧❡✐ss♥❡r✱ ▲✳ ❨❡♥❣✉❧❛❧♣✱ ✑❲❤❡♥ Cp ( X ) ✐s ❉♦♠❛✐♥ ❘❡♣r❡s❡♥t❛❜❧❡✑✱ ✷✵✶✸ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r② ❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐

❋✲❨ ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ s♣❛❝❡ ❲❡ 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❛❜❧❡✳ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r② ❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐

❚❤❡♦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 ❛ G δ ✕s✉❜s♣❛❝❡ ♦❢ X ✱ t❤❡♥ Y ✐s ❛ ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ s♣❛❝❡✳ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r② ❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐

❋✲❨ π ✕❞♦♠❛✐♥ r❡♣r❡s❡♥t❛❜❧❡ s♣❛❝❡ ❲❡ 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❛❜❧❡✳ ❚❤❡ ❇❛♥❛❝❤✕▼❛③✉r ❣❛♠❡ ❛♥❞ t❤❡ str♦♥❣ ❈❤♦q✉❡t ❣❛♠❡ ✐♥ ❞♦♠❛✐♥ t❤❡♦r② ❆♥❞r③❡❥ ❑✉❝❤❛rs❦✐