Về các tập w đóng suy rộng chính quy và các tập w nửa đóng suy rộng chính quy

▼ô❈ ▲ô❈ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✶ ▲ê✐ ♥ã✐ ➤➬✉ ✷ ❈❤➢➡♥❣ ✶✳ ❈➳❝ t❐♣ ω ✲➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ω ✲➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✹ ✶✳✶ ❚❐♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ω ✲T 12 ✶✳✸ ❍➭♠ r❣ω ✲❧✐➟♥ tô❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ❦❤➠♥❣ ❣✐❛♥ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ❈❤➢➡♥❣ ✷✳ ❈➳❝ t❐♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✷✳✶ ❚❐♣ ✷✳✷ ❈➳❝ ➳♥❤ ①➵ s✉② ré♥❣ tr➟♥ t❐♣ ✹ ✶✶ ✷✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✳ ✷✺ ❑Õt ❧✉❐♥ ✸✷ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ✸✸ ✶ ❧ê✐ ♥ã✐ ➤➬✉ ❈❤ó♥❣ t❛ ➤➲ ❧➭♠ q✉❡♥ ✈í✐ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ tÝ♥❤ ❝❤✃t t❤➢ê♥❣ ➤➢ỵ❝ sư ❞ơ♥❣ tr♦♥❣ ●✐➯✐ tÝ❝❤ ♥❤➢ t❐♣ ❤ỵ♣ ➤ã♥❣✱ t❐♣ ❤ỵ♣ ♠ë✱ ♣❤➬♥ tr♦♥❣ ❝đ❛ t❐♣ ❤ỵ♣✱ ❜❛♦ ➤ã♥❣ ❝đ❛ t❐♣ ❤ỵ♣✱ ✳✳✳ ▼ë ré♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ tÝ♥❤ ❝❤✃t ♥➭② ✈➭♦ ♥➝♠ ✶✾✼✵ ✈➭ ✶✾✽✷✱ ◆✳ ▲❡✈✐♥ ✈➭ ❍✳ ❩✳ ❍❞❡✐❜ ❧➬♥ ❧➢ỵt ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ t❐♣ ➤ã♥❣ s✉② ré♥❣ ✭❣✲ ➤ã♥❣✮ ✈➭ t❐♣ ω ✲➤ã♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✳ ❙❛✉ ➤ã ❝➳❝ ♥❤➭ t➠♣➠ ➤➲ ♠ë ré♥❣ ✈➭ ➤➢❛ r❛ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ♥❤➢ t❐♣ ♠ë ❝❤Ý♥❤ q✉②✱ t❐♣ ♥ư❛ ♠ë✱ t❐♣ ♥ư❛ t✐Ị♥ ♠ë✱ t❐♣ α✲♠ë✱ t❐♣ θ✲♠ë✱ t❐♣ δ ✲♠ë✱ ✳✳✳ ◆➝♠ ✶✾✽✼✱ P✳ ❇❛❤❛tt❛❝❤❛r②②❛ ✈➭ ❇✳ ❑✳ ▲❛❤✐r✐ ➤➲ ➤➢❛ r❛ ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ tÝ♥❤ ❝❤✃t ✈Ị ❝➳❝ t❐♣ s❣✲➤ã♥❣ ✈➭ s❣✲♠ë✳ ❈➳❝ t❐♣ s❣✲➤ã♥❣ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ♠ét ❝➳❝❤ ré♥❣ r➲✐ tr♦♥❣ ♥❤÷♥❣ ♥➝♠ ❣➬♥ ➤➞② ♣❤➬♥ ❧í♥ ❜ë✐ ❑✳ ❇❛❧❛❝❤❛♥❞r❛♥✱ ▼✳ ❈✳ ❈❛❧❞❛s✱ ❘✳ ❉❡✈✐✱ ❏✳ ❉♦♥t❝❤❡✈✱ ▼✳ ●❛♥st❡r✱ ❍✳ ▼❛❦✐✱ ❚✳ ◆♦✐r✐ ✈➭ P✳ ❙✉♥❞❛r❛♠✳ ◆➝♠ ✶✾✾✼✱ ❆✳ ❘❛♥✐ ✈➭ ❑✳ ❇❛❧❛❝❤❛♥❞r❛♥ ➤➲ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❝➳❝ t❐♣ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ s✉② ré♥❣ ❝❤Ý♥❤ q✉②✳ ➜Õ♥ ♥➝♠ ✷✵✵✼✱ ❆❤♠❛❞ ❆❧ ✲ ❖♠❛r✐ ✈➭ ▼♦❤❞ ❙❛❧♠✐ ▼❞ ◆♦♦r❛♥✐ ➤➲ ➤➢❛ r❛ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ♠í✐ ✈Ị t❐♣ ω ✲➤ã♥❣ s✉② ω ✲➤ã♥❣✮✱ ➳♥❤ ①➵ ω ✲❧✐➟♥ tô❝ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭r❣ω ✲❧✐➟♥ ré♥❣ ❝❤Ý♥❤ q✉② ✭r❣ tô❝✮✱ ➳♥❤ ①➵ ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭r❣ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✮ ✳✳✳ ❚r➟♥ ❝➡ së ❜➭✐ ❜➳♦ ❝ñ❛ ❆✳ ❘❛♥✐ ✈➭ ❑✳ ❇❛❧❛❝❤❛♥❞r❛♥ ✭✶✾✾✼✮✱ ❆❤♠❛❞ ❆❧ ✲ ❖♠❛r✐ ✈➭ ▼♦❤❞ ❙❛❧♠✐ ▼❞ ◆♦♦r❛♥✐ ✭✷✵✵✼✮ ✈➭ ❧✉❐♥ ✈➝♥ ❚❤➵❝ sÜ ❚♦➳♥ ❤ä❝ ❝đ❛ ◆❣✉②Ơ♥ ❚❤Þ ❚❤✉ ✭❈❛♦ ❤ä❝ ●✐➯✐ tÝ❝❤ ✶✹✮ ❝ï♥❣ ✈í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ t➳❝ ❣✐➯ ➤➲ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♥➭②✳ ▼ơ❝ ➤Ý❝❤ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ t❐♣ ❝❤Ý♥❤ q✉② ✭ ω ✲♥ư❛ ➤ã♥❣ s✉② ré♥❣ ω ✲sr❣✲➤ã♥❣✮✱ ①Ðt ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ♥ã ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ s✉② ré♥❣ tr➟♥ t❐♣ ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉②✱ ➤å♥❣ t❤ê✐ tr×♥❤ ❜➭② ❝ã ω ✲➤ã♥❣✱ ❧í♣ ❝➳❝ ❤Ư t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ❝➳❝ t❐♣ r❣ ❤➭♠ r❣ ω ✲❧✐➟♥ tô❝✱ ❤➭♠ r❣ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✳ ❱í✐ ♠ơ❝ ➤Ý❝❤ tr➟♥✱ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ tr×♥❤ ❜➭② t❤➭♥❤ ❤❛✐ ❝❤➢➡♥❣ ❈❤➢➡♥❣ ✶✳ ❈➳❝ t❐♣ ω ✲➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉②✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ❧í♣ ❝➳❝ t❐♣ q✉②✱ ω ✲T 12 ω ✲➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ ❦❤➠♥❣ ❣✐❛♥ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✈➭ ❧í♣ ❝➳❝ ➳♥❤ ①➵ ✷ ω ✲❧✐➟♥ tơ❝ s✉② ré♥❣ ❝❤Ý♥❤ q✉②✳ ❈❤➢➡♥❣ ✷✳ ❈➳❝ t❐♣ ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉②✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ t❐♣ ❝❤Ý♥❤ q✉② ✈➭ ❝➳❝ ➳♥❤ ①➵ s✉② ré♥❣ tr➟♥ t❐♣ ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ω ✲♥ư❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉②✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✳ ❚➳❝ ❣✐➯ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ s➞✉ s➽❝ ♥❤✃t ➤Õ♥ t❤➬②✳ ◆❤➞♥ ❞Þ♣ ♥➭②✱ t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❇❛♥ ❝❤đ ♥❤✐Ư♠ ❦❤♦❛ ❚♦➳♥✱ ❇❛♥ ❝❤đ ♥❤✐Ö♠ ❦❤♦❛ ❙❛✉ ➜➵✐ ❤ä❝✱ ❝➳❝ t❤➬② ❣✐➳♦✱ ❝➠ ❣✐➳♦ tr♦♥❣ ❦❤♦❛✱ ➤➷❝ ❜✐Öt ❧➭ ❝➳❝ t❤➬② ❝➠ ❣✐➳♦ tr♦♥❣ tæ ●✐➯✐ tÝ❝❤ ❦❤♦❛ ❚♦➳♥ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ ❣✐ó♣ ➤ì tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❚➳❝ ❣✐➯ ①✐♥ ❝➯♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❈❛♦ ❤ä❝ ❦❤♦➳ ✶✺ ●✐➯✐ tÝ❝❤ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❣✐ó♣ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ tr♦♥❣ s✉èt ❦❤♦➳ ❤ä❝✳ ▼➷❝ ❞ï ➤➲ ❝ã ♥❤✐Ị✉ ❝è ❣➽♥❣✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sót ú t rt ợ ữ ý ế ➤ã♥❣ ❣ã♣ ❝ñ❛ ❝➳❝ t❤➬② ❝➠ ❣✐➳♦ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥✳ ❱✐♥❤✱ t❤➳♥❣ ✶✷ ♥➝♠ ✷✵✵✾ ❚➳❝ ❣✐➯ ✸ ❝❤➢➡♥❣ ✶ ❈➳❝ t❐♣ ❚❐♣ ✶✳✶ ω ✲➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ω ✲➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ➜Þ♥❤ ♥❣❤Ü❛✳ ●✐➯ sư (X, τ ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ ✈➭ A ❧➭ t❐♣ ❝♦♥ ❝ñ❛ X ể xX ỗ A ợ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❝➠ ➤ä♥❣ ✭❝♦♥❞❡♥s❛t✐♦♥✮ ✭❬✹❪✮ ❝đ❛ U ∈τ ợ ọ xU tì A ế U A ❦❤➠♥❣ ➤Õ♠ ➤➢ỵ❝❀ ω ✲➤ã♥❣ ✭ω ✲❝❧♦s❡❞✮ ✭❬✶✵❪✮ ♥Õ✉ ♥ã ❝❤ø❛ t✃t ❝➯ ❝➳❝ ➤✐Ĩ♠ ❝➠ ➤ä♥❣ ❝đ❛ ♥ã✳ P❤➬♥ ❜ï ❝đ❛ t❐♣ ✶✳✶✳✷ ω ✲➤ã♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ω ✲♠ë ✭ω ✲♦♣❡♥✮✳ ▼Ư♥❤ ➤Ị✳ ✭❬✹❪✮ ❚❐♣ ❝♦♥ A ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ω ✲♠ë ế ỉ ế ỗ x A tồ t➵✐ t❐♣ U ∈ τ t❐♣ ➤➢ỵ❝ ❣ä✐ ❧➭ s❛♦ ❝❤♦ x∈U ✈➭ U − A ❧➭ t❐♣ ➤Õ♠ ➤➢ỵ❝✳ ❍ä t✃t ❝➯ ❝➳❝ t❐♣ ❝♦♥ t➠♣➠ tr➟♥ ✶✳✶✳✸ X ♠Þ♥ ❤➡♥ ω ✲♠ë ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ✭❳✱ τ ✮ ❦ý ❤✐Ư✉ ❜ë✐ τω ✱ ➤➞② ❧➭ τ✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❚❐♣ ❝♦♥ g ➤ã♥❣ s✉② ré♥❣ ✭ ✲➤ã♥❣✮ ♥Õ✉ P❤➬♥ ❜ï ❝ñ❛ t❐♣ ❚➠♣➠ ✶✳✶✳✹ ❝ñ❛ A ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ clA ⊆ U ✈í✐ ♠ä✐ t❐♣ ♠ë U ♠➭ A ⊆ U✳ g ✲➤ã♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ g ✲♠ë✳ τA ♥❤➽❝ ➤Õ♥ tr♦♥❣ ❧✉❐♥ ✈➝♥ ❝❤Ý♥❤ ❧➭ t➠♣➠ ❝➯♠ s✐♥❤ ❜ë✐ τ ❇ỉ ➤Ị✳ ✭❬✺❪✮ ●✐➯ sư (X, τ ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ ✈➭ X ✳ ❑❤✐ ➤ã ✭❛✮ (τω )ω = τω ❀ ✭❜✮ (τA )ω = (τω )A ✳ ✹ A tr➟♥ A✳ ❧➭ t❐♣ ❝♦♥ ω ✲❜❛♦ ➤ã♥❣ ✈➭ ω ✲♣❤➬♥ tr♦♥❣ ❝đ❛ t❐♣ A ➤Þ♥❤ ♥❣❤Ü❛ t➢➡♥❣ tù clA✱ intA ✈➭ ❝❤ó♥❣ ➤➢ỵ❝ ❦ý ❤✐Ư✉ ❧➬♥ ❧➢ỵt ❧➭ ✶✳✶✳✺ ◆❤❐♥ ①Ðt✳ ❈❤♦ clω (A)✱ intω (A)✳ A✱ B ❧➭ ❝➳❝ t❐♣ ❝♦♥ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ )✳ ❑❤✐ ➤ã ✭❛✮ ❍ỵ♣ ❝đ❛ ❤ä t✉ú ý ❝➳❝ t❐♣ ω ✲♠ë ❧➭ t❐♣ ω ✲♠ë✳ ❉♦ ➤ã intω (A) ❧➭ t❐♣ ω ✲♠ë❀ ✭❜✮ ●✐❛♦ ❝ñ❛ ❤ä tï② ý ❝➳❝ t❐♣ ω ✲➤ã♥❣ ❧➭ t❐♣ ω ✲➤ã♥❣✳ ❉♦ ➤ã clω (A) ❧➭ t❐♣ ω ✲➤ã♥❣❀ ✭❝✮ ◆Õ✉ A ❧➭ t❐♣ ♠ë✱ t❤× A ❧➭ t❐♣ ω ✲♠ë✳ ❉♦ ➤ã intA ⊂ intω (A)❀ ✭❞✮ ◆Õ✉ A ❧➭ t❐♣ ➤ã♥❣✱ t❤× A ❧➭ t❐♣ ω ✲➤ã♥❣✳ ❉♦ ➤ã clω (A) ⊂ clA❀ ✭❡✮ ◆Õ✉ A⊂B ✶✳✶✳✻ tì cl (A) cl (B) ị ĩ ❝♦♥ A ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ➤➢ỵ❝ ❣ä✐ ❧➭ ω ✲➤ã♥❣ s✉② ré♥❣ ✭❣❡♥❡r❛❧✐③❡❞ ω ✲❝❧♦s❡❞✮ ✈✐Õt ❣ä♥ ❧➭ ❣ω ✲➤ã♥❣ ♥Õ✉ clω (A) ⊆ U ✈í✐ ♠ä✐ ✶✳✶✳✼ U ∈τ ♠➭ A ⊆ U✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✻❪✮ ❚❐♣ ❝♦♥ ♠ë ❝❤Ý♥❤ q✉② ✭r❡❣✉❧❛r ♦♣❡♥✮ ♥Õ✉ A ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ➤➢ỵ❝ ❣ä✐ ❧➭ A = int(clA)✳ P❤➬♥ ❜ï ❝ñ❛ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② ✭r❡❣✉❧❛r ❝❧♦s❡❞✮✳ ▼ét ❝➳❝❤ t➢➡♥❣ ➤➢➡♥❣✱ t❐♣ ✶✳✶✳✽ A ❧➭ ➤ã♥❣ ❝❤Ý♥❤ q✉② ế A = cl(intA) ét ỗ t í q ỗ t ó í q ó ✶✳✶✳✾ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✹❪✮ ❚❐♣ ❝♦♥ A ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭r❡❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ❝❧♦s❡❞✮ ✈✐Õt ❣ä♥ ❧➭ r❣✲➤ã♥❣ ♥Õ✉ clA ⊂ U ✈í✐ ♠ä✐ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ✺ U ♠➭ A ⊂ U✳ ✶✳✶✳✶✵ ❧➭ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✸❪✮ ❚❐♣ ❝♦♥ A ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ➤➢ỵ❝ ❣ä✐ ω ✲➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭r❡❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ω ✲❝❧♦s❡❞✮ ✈✐Õt ❣ä♥ ❧➭ r❣ω ✲ ➤ã♥❣ ♥Õ✉ clω (A) ⊂ U P❤➬♥ ❜ï ❝đ❛ t❐♣ ✈í✐ ♠ä✐ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ω ✲➤ã♥❣ U ♠➭ A ⊂ U✳ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ω ✲♠ë s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭r❡❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ω ✲♦♣❡♥✮ ✈✐Õt ❣ä♥ ❧➭ r❣ω ✲♠ë✳ ✶✳✶✳✶✶ A ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, ) ợ ọ ị ĩ ω ✲❝✲➤ã♥❣ ✭ω ✲❝✲❝❧♦s❡❞✮ ♥Õ✉ tå♥ t➵✐ t❐♣ B ⊂ A s❛♦ ❝❤♦ A = clω (B)✳ ❚õ ❝➳❝ ➤Þ♥❤ ♥❣❤Ü❛ ✈➭ tÝ♥❤ ❝❤✃t ➤➲ ❜✐Õt t❛ ❝ã t❤Ó ❦Õt ợ ố ệ ữ t r ó ✈í✐ ❝➳❝ t❐♣ ❦❤➳❝ ➤➲ ❜✐Õt q✉❛ s➡ ➤å s❛✉ ω − ❝ − ➤ã♥❣ ⇑ ⇒ ➤ã♥❣ g − ➤ã♥❣ ⇓ ⇓ ω − ➤ã♥❣ ✶✳✶✳✶✷ ❱Ý ❞ô✳ ✭❬✶✸❪✮ ❳Ðt tû ✈í✐ t➠♣➠ ❞♦ rg − ➤ã♥❣ ⇓ ⇒ gω − ➤ã♥❣ ⇒ rgω − ➤ã♥❣ R ❧➭ t❐♣ t✃t ❝➯ ❝➳❝ sè t❤ù❝✱ Q ❧➭ t❐♣ t✃t ❝➯ ❝➳❝ sè ❤÷✉ τ = {∅, R, R − Q}✳ ❚❤❐t ✈❐②✱ ❞♦ ⇒ ❑❤✐ ➤ã A = R − Q ❦❤➠♥❣ ❧➭ t❐♣ ❣ω ✲➤ã♥❣✳ A ❧➭ t❐♣ ♠ë ♥➟♥ A ❝ò♥❣ ❧➭ ω ✲♠ë ✈➭ A ⊆ A✱ ♥❤➢♥❣ clω (A) ⊆ A✱ A ❦❤➠♥❣ ❧➭ t❐♣ ω ✲➤ã♥❣✳ ❱❐② A ❦❤➠♥❣ ❧➭ t❐♣ ❣ω ✲➤ã♥❣✳ ▼➷t ❦❤➳❝ ❝❤Ø ❝ã ♠ét t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❝❤ø❛ ✶✳✶✳✶✸ A ❧➭ R✱ ❞♦ ➤ã A ❧➭ t❐♣ r❣ω ✲➤ã♥❣✳ ❱Ý ❞ô✳ ✭❬✶✸❪✮ ❈❤♦ X = {a, b, c, d} {∅, X, {a}, {b}, {a, b}, {a, b, c}}✳ {❛} ❧➭ t❐♣ r❣ω ✲➤ã♥❣✱ ❞♦ X ✶✳✶✳✶✹ ❑❤✐ ➤ã ❤÷✉ ❤➵♥ ✈➭ τω ✈í✐ t➠♣➠ ➤➢ỵ❝ ❝❤♦ ❜ë✐ ❤ä {❛} ❦❤➠♥❣ ❧➭ t❐♣ r❣✲➤ã♥❣✳ τ = ◆❤➢♥❣ ❧➭ t➠♣➠ rê✐ r➵❝✳ ị ý ỗ t ó t ró ❧➭ t❐♣ r❣ω ✲➤ã♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö q✉② ❜✃t ❦ú ❝❤ø❛ A✳ A ❧➭ t❐♣ ❣ω ✲➤ã♥❣ tr♦♥❣ (X, τ ) ✈➭ U ❚õ ◆❤❐♥ ①Ðt ✶✳✶✳✽ s✉② r❛ clω (A) ⊂ U ✳ ❉♦ ➤ã A ❧➭ t❐♣ r❣ω ✲➤ã♥❣✳ ✻ U ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ ❧➭ t❐♣ ♠ë ❝❤ø❛ A✱ ♥➟♥ t❛ ❝ã A ❧➭ t❐♣ r❣✲➤ã♥❣ tr♦♥❣ (X, τ ) ✈➭ U ◆Õ✉ ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❝❤ø❛ A✳ ❑❤✐ ➤ã t❛ ❝ã clA ⊂ U ✳ ▼➷t ❦❤➳❝ t❛ ❧✉➠♥ ❝ã clω (A) ⊂ clA ❞♦ ➤ã clω (A) ⊂ U ✳ ❱❐② ❆ t❐♣ r❣ ω ✲➤ã♥❣✳ ✶✳✶✳✶✺ ➜Þ♥❤ ❧ý✳ ❈❤♦ A ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝ñ❛ (X, τ )✳ ❑❤✐ ➤ã clω (A) − A X✳ ❦❤➠♥❣ ❝❤ø❛ ❜✃t ❦ú t❐♣ ó í q rỗ ủ ứ sö F ⊆ clω (A) − A✳ r❣ ❑❤✐ ➤ã ω ✲➤ã♥❣ ✈➭ X − F ❦Ð♦ t❤❡♦ ❧➭ t❐♣ ❝♦♥ ➤ã♥❣ ❝❤Ý♥❤ q✉② ❝ñ❛ ❧➭ t❐♣ ❝♦♥ ♠ë ❝❤Ý♥❤ q✉② ❝đ❛ ❱× t❤Õ (X, τ ) ♥➟♥ clω (A) ⊆ X − F F ⊆ clω (A) ∩ (X − clω (A)) = ∅✳ A (X, τ ) ❝ñ❛ ω ✲♠ë ❧➭ r❣ intω (A) ✈í✐ ♠ä✐ t❐♣ ❝♦♥ ➤ã♥❣ ❝❤Ý♥❤ q✉② F ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö ❝❤Ý♥❤ q✉② ❜✃t ❦ú ❝đ❛ ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ ♠➭ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ➜✐Ò✉ (X, τ ) ♠➭ F ⊆ A✳ ❑❤✐ ➤ã X−A ❧➭ t❐♣ ❝♦♥ ➤ã♥❣ ❧➭ t❐♣ r❣ ω ✲➤ã♥❣✱ X − A ⊆ X − F ✳ ❉♦ ➤ã t❛ ❝ã X − intω (A) = ❦Ð♦ t❤❡♦ F ⊆ intω (A)✳ F ⊆ intω (A) ✈í✐ ♠ä✐ t❐♣ ❝♦♥ ➤ã♥❣ ❝❤Ý♥❤ q✉② ❧➭ t❐♣ ❝♦♥ ♠ë ❝❤Ý♥❤ q✉② ❜✃t ❦ú ♠➭ t❐♣ ❝♦♥ ➤ã♥❣ ❝❤Ý♥❤ q✉② ♠➭ F ⊆ F ⊆ A✳ A ❧➭ t❐♣ ❝♦♥ r❣ω ✲♠ë ❝ñ❛ (X, τ )✱ F ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ✈➭ clω (X − A) ⊆ X − F X −A⊆U t❛ ❝ã ♥➟♥ F ♠➭ X −U X − U ⊆ A✱ ❞♦ ➤ã X − U ⊆ intω (A)✳ X − intω (A) = clω (X − A) ⊆ U r❣ A ❧➭ t❐♣ X✳ ➜Þ♥❤ ❧ý✳ ❚❐♣ ❝♦♥ F ⊆ A ✈➭ U ❱× s❛♦ ❝❤♦ F = ∅✳ ❱❐② clω (A) − A ❦❤➠♥❣ ❝❤ø❛ t ỳ t ó í q rỗ ủ X −F (X, τ ) F ⊆ X − A ✈➭ ❞♦ ➤ã A ⊆ X − F ✳ F ⊆ X − clω (A)✳ ♥➭② ❦Ð♦ t❤❡♦ ✶✳✶✳✶✻ F ❧➭ ❙✉② r❛ X − A ❧➭ t❐♣ r❣ω ✲➤ã♥❣✳ ❱❐② A ❧➭ t❐♣ ω ✲♠ë✳ ✶✳✶✳✶✼ ♠ä✐ ❇ỉ ➤Ị✳ ✭❬✽❪✮ ❱í✐ ♠ä✐ t❐♣ ♠ë A⊆X ✶✳✶✳✶✽ t❛ ❝ã U ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ✈➭ ✈í✐ cl(U ∩ A) = cl(U ∩ clA)✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❍❛✐ t❐♣ rỗ A B ợ ọ t ợ ế clA∩ B = A ∩ clB = ∅✳ ✼ ✶✳✶✳✶✾ ❱Ý ❞ơ✳ ✭❬✶✸❪✮ ❈❤♦ ❝♦♥ ❦❤➠♥❣ ➤Õ♠ ➤➢ỵ❝ ❝đ❛ ✈í✐ t➠♣➠ X ❧➭ t❐♣ ❦❤➠♥❣ ➤Õ♠ ➤➢ỵ❝ ✈➭ A✱ B ✱ C ✱ D ❧➭ ❝➳❝ t❐♣ X ✈➭ ❤ä {A, B, C, D} ❧➭ ♠ét sù ♣❤➞♥ ❤♦➵❝❤ ❝ñ❛ X τ = {∅, X, A, B, A ∪ B, A ∪ B ∪ C}✳ ❈❤ä♥ ❝➳❝ t❐♣ r❣ x, y ∈ / A ✈➭ x = y✱ ❦❤✐ ➤ã H = A ∪ {x} ✈➭ G = A ∪ {y} ω ✲➤ã♥❣ ❞♦ ❝❤Ø ❝ã ♠ét t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❝❤ø❛ H ✱ G ❧➭ X ✳ H∩G = A ✈➭ A ♠ë ❝❤Ý♥❤ q✉② tr♦♥❣ X ✱ clω (A) ⊆ A ✈× A ❧➭ ◆❤➢♥❣ ❦❤➠♥❣ ❧➭ t❐♣ ω ✲➤ã♥❣ ❦Ð♦ t❤❡♦ H ∩ G ❦❤➠♥❣ ❧➭ t❐♣ r❣ω ✲➤ã♥❣✳ ω ✲♠ë ❦❤➠♥❣ ❧➭ t❐♣ r❣ω ✲♠ë✳ ❉♦ ➤ã ❤ỵ♣ ❝đ❛ ❝➳❝ t❐♣ r❣ ✶✳✶✳✷✵ r❣ ❧➭ ❝➳❝ t❐♣ r❣ ω ✲➤ã♥❣✱ t❤× A ∪ B ❧➭ t❐♣ ω ✲➤ã♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö A⊂U r❛ A ✈➭ B ➜Þ♥❤ ❧ý✳ ✭❬✶✸❪✮ ◆Õ✉ ✈➭ U ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② s❛♦ ❝❤♦ B ⊂ U ✳ ❱× A✱ B ❧➭ r❣ A ∪ B ⊂ U ✳ ❑❤✐ ➤ã ω ✲➤ã♥❣ ♥➟♥ clω (A) ⊂ U ✱ clω (B) ⊂ U ✳ ❙✉② clω (A ∪ B) = clω (A) ∪ clω (B) ⊂ U ✳ ❱❐② A ∪ B ❧➭ r❣ω ✲➤ã♥❣✳ A ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝ñ❛ (X, τ )✳ ◆Õ✉ B ⊆ X ✶✳✶✳✷✶ ➜Þ♥❤ ❧ý✳ ✭❬✶✸❪✮ ❈❤♦ s❛♦ ❝❤♦ A ⊆ B ⊆ clω (A)✱ t❤× B ◆Õ✉ ω ✲➤ã♥❣✳ ❝ị♥❣ ❧➭ t❐♣ r❣ B ❧➭ t❐♣ ❝♦♥ ❝ñ❛ (X, τ ) ✈➭ A ❧➭ t❐♣ ❝♦♥ r❣ω ✲♠ë s❛♦ ❝❤♦ intω (A) ⊆ B ⊆ A t❤× B ω ✲♠ë✳ ❝ị♥❣ ❧➭ t❐♣ r❣ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö B ⊆ clω (A)✳ A ❑❤✐ ➤ã t❛ ❝ã clω (A) = clω (B)✳ ◆Õ✉ U ❧➭ t❐♣ ❝♦♥ r❣ ω ✲➤ã♥❣ ❝ñ❛ (X, τ ) s❛♦ ❝❤♦ A ⊆ clω (A) ⊆ clω (B) ⊆ clω (clω (A)) = clω (A) ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❜✃t ❦ú ❝❤ø❛ B t❤× U ♥➟♥ ❝ò♥❣ ❝❤ø❛ A ♥➟♥ clω (A) = clω (B) ⊆ U ✳ ❱❐② B ❧➭ t❐♣ r❣ω ✲➤ã♥❣✳ P❤➬♥ ❝ß♥ ❧➵✐ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù✳ ✶✳✶✳✷✷ t❐♣ r❣ ➜Þ♥❤ ❧ý✳ ◆Õ✉ A ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝ñ❛ (X, τ )✱ t❤× clω (A) − A ❧➭ ω ✲♠ë✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö ➤ã♥❣ ❝❤Ý♥❤ q✉② s❛♦ ❝❤♦ r❛ F = ∅ ✈➭ A ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝ñ❛ (X, τ ) ✈➭ F F ⊆ clω (A) − A✳ F ⊆ intω (clω (A) − A)✳ clω (A) − A ❧➭ t❐♣ r❣ω ✲♠ë✳ ✽ ❧➭ t❐♣ ❝♦♥ ó từ ị ý t s ì t❤❡♦ ➜Þ♥❤ ❧ý ✶✳✶✳✶✻ t❛ s✉② r❛ ✶✳✶✳✷✸ Y ❇ỉ ➤Ị✳ ✭❬✹❪✮ ◆Õ✉ ❧➭ t❐♣ ❝♦♥ ❝đ❛ ✶✳✶✳✷✹ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ♠ë ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ X ✈➭ A Y ✱ t❤× clω|Y (A) = clω (A) ∩ Y ✳ ❇ỉ ➤Ị✳ ✭❬✶✸❪✮ ◆Õ✉ A ω ✲➤ã♥❣ ❧➭ t❐♣ ❝♦♥ ♠ë ❝❤Ý♥❤ q✉② ✈➭ r❣ ❝đ❛ (X, τ )✱ t❤× A ❧➭ t❐♣ ω ✲➤ã♥❣ tr♦♥❣ X ✳ ❈❤ø♥❣ ♠✐♥❤✳ ➜Ó ❝❤ø♥❣ ♠✐♥❤ ❚❤❐t ✈❐②✱ ❞♦ A ❧➭ t❐♣ ω ✲➤ã♥❣ t❛ ❝❤ø♥❣ ♠✐♥❤ clω (A) = A✳ A ♠ë ❝❤Ý♥❤ q✉②✱ A ⊆ A ✈➭ A ❧➵✐ ❧➭ t❐♣ r❣ω ✲➤ã♥❣ ♥➟♥ clω (A) ⊆ A✳ ▼➭ A ⊆ clω (A)✳ ❙✉② r❛ clω (A) = A✳ ❱❐② A ❧➭ t❐♣ ω ✲➤ã♥❣ tr♦♥❣ X ✳ ➜Þ♥❤ ❧ý✳ ❈❤♦ Y ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ♠ë ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ X ✈➭ A ✶✳✶✳✷✺ ◆Õ✉ A ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ X ✱ t❤× A ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ Y ✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö U = V ∩ Y ✱ ✈í✐ V ♥➟♥ U ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❝ñ❛ Y ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❝ñ❛ s❛♦ ❝❤♦ A ⊆ U ✳ ❑❤✐ ➤ã X ✳ ❉♦ A ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ X clω (A) ⊆ V ✳ ◆❤ê ❇ỉ ➤Ị ✶✳✶✳✷✸ t❛ ❝ã clω|Y (A) = clω (A)∩Y ⊆ V ∩Y = U ✳ ❉♦ ➤ã A ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ Y ✳ ✶✳✶✳✷✻ ❍Ö q✉➯✳ ✭❬✶✸❪✮ ◆Õ✉ A ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉②✱ ω ✲➤ã♥❣ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ X ✱ t❤× A ∩ B ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö ❝❤ø♥❣ ♠✐♥❤ ❣✐➯ t❤✐Õt t❛ ❝ã ❉♦ U ❧➭ t❐♣ r❣ r❣ ω ✲➤ã♥❣ ✈➭ B ❧➭ t❐♣ A∩B ⊆ U t❛ ❝➬♥ ω ✲➤ã♥❣✳ ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② s❛♦ ❝❤♦ clω (A ∩ B) ⊆ U ✳ ❚❤❐t ✈❐②✱ ✈× A ♠ë ❝❤Ý♥❤ q✉② ✈➭ r❣ω ✲➤ã♥❣ ♥➟♥ t❤❡♦ ❇ỉ ➤Ị ✶✳✶✳✷✹ t❛ ❝ã U✳ ⊆Y✳ A ❧➭ t❐♣ ω ✲➤ã♥❣✳ ❑Ð♦ t❤❡♦ clω (A) = A✳ ➜å♥❣ t❤ê✐ t❤❡♦ clω (B) = B ✳ ▼➷t ❦❤➳❝ A ❧➭ r❣ω ✲➤ã♥❣ ✈➭ A ⊂ U ✱ ♥➟♥ clω (A) ⊂ A ∩ B ⊆ A ♥➟♥ clω (A ∩ B) ⊆ clω (A)✳ ▼➷t ❦❤➳❝ A∩B ⊆ B ♥➟♥ clω (A ∩ B) ⊆ clω (B)✳ ❙✉② r❛ clω (A ∩ B) ⊆ clω (A) ∩ clω (B) ⊆ clω (A) ⊂ U ✳ ❱❐② A ∩ B ❧➭ t❐♣ r❣ω ✲➤ã♥❣✳ A ❧➭ t❐♣ r❣ω ✲➤ã♥❣✳ ❑❤✐ ➤ã A = clω (intω (A)) ♥Õ✉ ✈➭ ✶✳✶✳✷✼ ➜Þ♥❤ ❧ý✳ ❈❤♦ ❝❤Ø ♥Õ✉ clω (intω (A)) − A ❧➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉②✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö A ❧➭ t❐♣ r❣ ω ✲➤ã♥❣✳ ◆Õ✉ A = clω (intω (A)) clω (intω (A)) − A = ∅ ❞♦ ➤ã clω (intω (A)) − A ➤ã♥❣ ❝❤Ý♥❤ q tì ợ sử ó í q clω (intω (A))−A ➤ã♥❣ ❝❤Ý♥❤ q✉②✳ ❱× clω (A)−A ❝❤ø❛ t❐♣ clω (intω (A))−A✱ ♥➟♥ ♥❤ê ➜Þ♥❤ ❧ý ✶✳✶✳✶✺ t❛ ❝ã clω (intω (A))− A = ∅✳ ❉♦ ➤ã A = clω (intω (A))✳ ✶✳✶✳✷✽ ❇ỉ ➤Ị✳ ✭❬✹❪✮ ❈❤♦ (X, τ ) ✈➭ (Y, σ) ❧➭ ❤❛✐ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✳ ❑❤✐ ➤ã (τ × σ)ω ⊆ τω × σω ✳ ị ý ế AìB t r ✲♠ë ❝ñ❛ (X, τ ) ✈➭ B ω ✲♠ë ❝ñ❛ (Y, σ)✳ ❧➭ t❐♣ ❝♦♥ r❣ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö B ❧➭ t❐♣ ❝♦♥ ➤ã♥❣ ❝❤Ý♥❤ q✉② ❝đ❛ ω ✲♠ë ❝đ❛ (X × Y, τ × σ) t❤× A ❧➭ t❐♣ ❝♦♥ r❣ ❧➭ t❐♣ ❝♦♥ r❣ (X, τ ) ✈➭ FB (Y, σ) s❛♦ ❝❤♦ FA ⊆ A✱ FB ⊆ B ✳ ❚❤Õ t❤× ω ✲♠ë ❝đ❛ (X × Y, τ × σ)✱ FA ❧➭ t❐♣ ❝♦♥ ➤ã♥❣ ❝❤Ý♥❤ q✉② ❝đ❛ FA × FB tr♦♥❣ (X × Y, τ × σ) t❐♣ r❣ ω ✲♠ë tr♦♥❣ (X × Y, τ × σ) ✈➭ tõ ❇ỉ ➤Ị ✶✳✶✳✷✽ t❛ s✉② r❛ FA × FB ⊆ s❛♦ ❝❤♦ FA × FB ⊆ A × B ✳ ❧➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② intω (A × B) ⊆ intω (A) × intω (B)✳ ❉♦ ➤ã ❱× ✈❐② ❚õ ❣✐➯ t❤✐Õt A×B ❧➭ FA ⊆ intω (A)✱ FB ⊆ intω (B)✳ A✱ B ❧➭ t r ề ợ ủ ị ý tr➟♥ ❦❤➠♥❣ ➤ó♥❣✱ t❤Ĩ ❤✐Ư♥ q✉❛ ❱Ý ❞ơ s❛✉ X = Y = R ✈í✐ t➠♣➠ t❤➠♥❣ t❤➢ê♥❣ τ ✈➭ A = √ {{R − Q} ∪ [ 2; 5]}✱ B = (1; 7)✳ ❑❤✐ ➤ã A ✈➭ B ❧➭ ❝➳❝ t❐♣ ❝♦♥ r❣ω ✲♠ë ✭ω ✲ ✶✳✶✳✸✵ ❱Ý ❞ô✳ ✭❬✶✸❪✮ ❈❤♦ (R, τ )✱ A × B ❦❤➠♥❣ ❧➭ t❐♣ r❣ω ✲♠ë tr♦♥❣ (R × R, τ × τ )✱ ❞♦ t❐♣ F = √ [ 2; 3] × [3; 5] ❧➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② ❝❤ø❛ tr♦♥❣ A × B ✈➭ F ⊆ intω (A × B)✳ √ √ √ ➜✐Ó♠ ( 2; 4) ∈ F ✈➭ ( 2; 4) ∈ / intω (A × B)✱ ✈× ♥Õ✉ ( 2; 4) ∈ intω (A × B)✱ √ t❤× tå♥ t➵✐ t❐♣ ♠ë U ❝❤ø❛ ✈➭ t❐♣ ♠ë V ❝❤ø❛ ✹ s❛♦ ❝❤♦ (U × V ) − (A ì B) ủ t ế ợ t❐♣ ♠ë U √ ❝❤ø❛ (U × V ) − (A ì B) t ế ợ ọ ✈➭ ♠ä✐ t❐♣ ♠ë V ❝❤ø❛ ✹✳ ✶✵ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ✈➭ A ⊆ f −1 (U )✳ ▲➵✐ ❞♦ A ❧➭ t❐♣ r❣ω ✲➤ã♥❣ ♥➟♥ clω (A) ⊆ f −1 (U )✳ ❉♦ ➤ã f (clω (A)) ⊆ U ✳ ❉♦ f ❧➭ t✐Ò♥ ω ✲➤ã♥❣ s✉② r❛ f (clω (A)) ❧➭ t❐♣ ω ✲ clω (f (clω (A))) = f (clω (A))✳ ❱× ✈❐② clω (f (A)) ⊆ clω (f (clω (A))) ⊆ ➤ã♥❣ ✈➭ U ✳ ❉♦ ➤ã f (A) ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ Y ✳ ✶✳✸✳✶✼ ➜Þ♥❤ ❧ý✳ ✭❬✶✸❪✮ ❈❤♦ ➳♥❤ ①➵ ω ✲❧✐➟♥ tô❝✳ ❑❤✐ ➤ã f ❜➯♦ t♦➭♥ ✈➭ r❣ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư ♠ë ❝❤Ý♥❤ q✉② ❝đ❛ t❛ ❝ã U ❧➭ t❐♣ ❝♦♥ ❧➭ ➳♥❤ ①➵ f (U ) ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❝ñ❛ Y ✳ ◆❤ê ❣✐➯ t❤✐Õt V ❱× f ❧➭ t❐♣ ω ✲❧✐➟♥ tơ❝ ❧➭ r❣ clω (f −1 (V )) ⊆ U ✳ ❱× ✈❐② f −1 (V ) ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝ñ❛ X ✳ ❉♦ ➤ã f ✶✳✸✳✶✽ ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✳ ị ý ỗ t r cl (f −1 (clω (V ))) ⊆ U ✳ ❱× clω (f −1 (V )) ⊆ clω (f −1 (clω (V ))) ⊆ U ✱ ❧➭ ➳♥❤ ①➵ r❣ ω ✲♠ë V ❝ñ❛ Y ➳♥❤ ①➵ f B ❝ñ❛ Y s❛♦ ❝❤♦ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö ❞♦ ➤ã t❐♣ r❣ X : (X, τ ) −→ (Y, σ) ❧➭ r❣ω ✲➤ã♥❣ ♥Õ✉ ỉ ế ỗ t BV f ❝❤ø❛ f −1 (B)✱ tå♥ t➵✐ ♠ét t❐♣ f −1 (V ) ⊆ U ✳ ❧➭ ➳♥❤ ①➵ r❣ f −1 (B) ⊆ U ✳ U ω ✲➤ã♥❣✱ B ❧➭ t❐♣ ❝♦♥ ❝ñ❛ X −U Y ✈➭ U ❧➭ X ✈➭ f (X − U ) ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ Y ✳ ➜➷t V = Y − f (X − U ) t❤× V ❧➭ t❐♣ ♠ë ❝đ❛ s❛♦ ❝❤♦ ❑❤✐ ➤ã ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ ω ✲♠ë ✈➭ f −1 (V ) = f −1 (Y − f (X − U )) = X − (X − U ) = U ✳ ❱× ✈❐② ❧➭ t❐♣ r❣ ω ✲♠ë ❝❤ø❛ B s❛♦ ❝❤♦ f −1 (V ) ⊆ U ✳ ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư r➺♥❣ X −F Y ω ✲➤ã♥❣ ❜✃t ❦ú ❝ñ❛ Y ❧➭ t❐♣ ❝♦♥ r❣ clω (V ) ❧➭ ω ✲➤ã♥❣ tr♦♥❣ Y ✱ ❞♦ ➤ã f −1 (clω (V )) ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝đ❛ X ✈➭ ✈× ✈❐② V ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✳ ω ✲➤ã♥❣✱ ❞♦ ➤ã clω (V ) ⊆ f (U ) ✈➭ f −1 (clω (V )) ⊆ U ✳ ✈➭ r♦✲ ❧➭ ➳♥❤ ①➵ r❣ X s❛♦ ❝❤♦ f −1 (V ) ⊆ U ✳ ❘â r➭♥❣ V ⊆ f (U )✳ ❱× f r♦✲❜➯♦ t♦➭♥ t❛ s✉② r❛ r❣ V f : (X, τ ) −→ (Y, σ) ❧➭ s♦♥❣ ➳♥❤✱ ✈➭ s❛♦ ❝❤♦ ❦Ð♦ t❤❡♦ ❧➭ t❐♣ r❣ X −F F ❧➭ ♠ét t❐♣ ➤ã♥❣ ❝ñ❛ X ✳ ❑❤✐ ➤ã f −1 (Y −f (F )) ⊆ ❧➭ t❐♣ ♠ë✳ ❚õ ❣✐➯ t❤✐Õt s✉② r❛ tå♥ t➵✐ ♠ét t❐♣ r❣ Y − f (F ) ⊆ V ✈➭ f −1 (V ) ⊆ X − F ✳ ❉♦ ➤ã ω ✲♠ë V ❝ñ❛ F ⊆ X − f −1 (V )✱ Y − V ⊆ f (F ) ⊆ f (X − f −1 (V )) ⊆ Y − V ✳ ❙✉② r❛ f (F ) = Y − V ω ✲➤ã♥❣✳ ❱❐② f ❧➭ r❣ ω ✲➤ã♥❣✳ ✷✵ ❝❤➢➡♥❣ ✷ ❝➳❝ t❐♣ t❐♣ ✷✳✶ ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ➜Þ♥❤ ♥❣❤Ü❛✳ ●✐➯ sư (X, τ ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ ✈➭ A ❧➭ t❐♣ ❝♦♥ ❝đ❛ ✷✳✶✳✶ X✳ ✭❛✮ A ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ♥ư❛ ♠ë ✭s❡♠✐ ♦♣❡♥✮ ♥Õ✉ tå♥ t➵✐ t❐♣ ♠ë V s❛♦ ❝❤♦ V ⊆ A ⊆ clV ❀ ✭❜✮ A ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ♥ö❛ ➤ã♥❣ ✭s❡♠✐ ❝❧♦s❡❞✮ ♥Õ✉ X − A ❧➭ t❐♣ ♥ư❛ ♠ë❀ ✭❝✮ A ➤➢ỵ❝ ❣ä✐ ❧➭ ω ✲♥ö❛ ω ✲s❡♠✐ ♠ë ✭ ♦♣❡♥✮ ♥Õ✉ tå♥ t➵✐ t❐♣ ♠ë V s❛♦ ❝❤♦ V ⊆ A ⊆ clω (V )❀ ✭❞✮ A ➤➢ỵ❝ ❣ä✐ ❧➭ ω ✲♥ư❛ ➤ã♥❣ ✭ω ✲s❡♠✐ ❝❧♦s❡❞ ✮ ♥Õ✉ X − A ❧➭ t❐♣ ω ✲♥ö❛ ♠ë✳ ❚❐♣ t✃t ❝➯ ❝➳❝ t❐♣ ω ✲♥ö❛ ♠ë ❝đ❛ X ❚❐♣ t✃t ❝➯ ❝➳❝ t❐♣ ω ✲♥ư❛ ➤ã♥❣ ❝đ❛ X ❍ỵ♣ ❝đ❛ t✃t ❝➯ ❝➳❝ t❐♣ ❦ý ❤✐Ư✉ ❧➭ ωSO(X)✳ ❦ý ❤✐Ư✉ ❧➭ ωSC(X)✳ ω ✲♥ư❛ ♠ë ♥➺♠ tr♦♥❣ A ➤➢ỵ❝ ❣ä✐ ❧➭ ω ✲♥ư❛ ♣❤➬♥ ω ✲s❡♠✐ ✐♥t❡r✐♦r✮ ❝đ❛ A ❦ý ❤✐Ư✉ ❧➭ sintω (A)✳ tr♦♥❣ ✭ ●✐❛♦ ❝đ❛ t✃t ❝➯ ❝➳❝ t❐♣ ω ✲♥ư❛ ➤ã♥❣ ❝❤ø❛ A ➤➢ỵ❝ ❣ä✐ ❧➭ ω ✲♥ư❛ ❜❛♦ ➤ã♥❣ ω ✲s❡♠✐ ❝❧♦s✉r❡✮ ❝đ❛ A ❦ý ❤✐Ư✉ ❧➭ sclω (A)✳ ✭ ●✐❛♦ ❝đ❛ t✃t ❝➯ ❝➳❝ t❐♣ ♥ư❛ ➤ã♥❣ ❝❤ø❛ ✭s❡♠✐ ❝❧♦s✉r❡✮ ❝đ❛ ✷✳✶✳✷ A ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❜❛♦ ➤ã♥❣ A ý ệ scl(A) ét ỗ t t ỗ t ✲♥ư❛ ♠ë ❧➭ t❐♣ ♥ư❛ ♠ë❀ ✭❜✮ ❍ỵ♣ ❝đ❛ ❤ä t✉ú ý ❝➳❝ t❐♣ ❧➭ t❐♣ ω ✲♥ö❛ ♠ë ❧➭ t❐♣ ω ✲♥ö❛ ♠ë✳ ❉♦ ➤ã sintω (A) ω ✲♥ö❛ ♠ë❀ ✷✶ ✭❝✮ ●✐❛♦ ❝ñ❛ ❤ä tï② ý ❝➳❝ t❐♣ ω ✲♥ö❛ ➤ã♥❣ ❧➭ t❐♣ ω ✲♥ö❛ ➤ã♥❣✳ ❉♦ ➤ã sclω (A) ❧➭ t❐♣ ω ✲♥ö❛ ➤ã♥❣❀ X − sclω (U ) = sintω (X − U )❀ ✭❞✮ ✭❡✮ ◆Õ✉ ✭❢✮ A ⊂ B ✱ t❤× sclω (A) ⊂ sclω (B) ✈➭ sintω (A) ⊂ sintω (B)❀ scl(A) ⊂ sclω (A)❀ ✭❣✮ A ❧➭ t❐♣ ω ✲♥ö❛ ➤ã♥❣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ A = sclω (A)✳ ✷✳✶✳✸ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✷❪✮ ❚❐♣ ❝♦♥ ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ✭ ω ✲❣❡♥❡r❛❧✐③❡❞ sclω (A) ⊂ U ✱ ✈í✐ ♠ä✐ t❐♣ ♠ë U P❤➬♥ ❜ï ❝ñ❛ t❐♣ ❚❐♣ t✃t ❝➯ ❝➳❝ t❐♣ ✭t➢➡♥❣ ø♥❣✱ ♠➭ s❡♠✐ ❝❧♦s❡❞✮ ✈➭ ✈✐Õt ❧➭ ω ❣s✲➤ã♥❣ ♥Õ✉ A ⊂ U✳ ω ❣s✲➤ã♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ω ✲♥ö❛ ♠ë s✉② ré♥❣ ✭ω ✲❣❡♥❡r❛❧✐③❡❞ s❡♠✐ ♦♣❡♥✮ ✈➭ ✈✐Õt ❧➭ ✷✳✶✳✹ A ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ➤➢ỵ❝ ❣ä✐ ❧➭ ω ❣s✲♠ë✳ ω ❣s✲➤ã♥❣ ✭ω ❣s✲♠ë✮ tr♦♥❣ X ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ωGSC(X, τ ) ωGSO(X, τ )✮✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❚❐♣ ❝♦♥ A ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ➤➢ỵ❝ ❣ä✐ ❧➭ ω ✲♥ư❛ ω ✲s❡♠✐ r❡❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ❝❧♦s❡❞✮ ♥Õ✉ sclω (A) ⊂ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭ U ✱ ✈í✐ ♠ä✐ t❐♣ ♠ë ❝❤Ý♥❤ q✉② U ❚❐♣ A ➤➢ỵ❝ ❣ä✐ ❧➭ ❡r❛❧✐③❡❞ ♦♣❡♥✮ ♥Õ✉ ω ✲♥ö❛ ♠➭ A⊂U ω ✲sr❣✲➤ã♥❣✳ ✈➭ ✈✐Õt ❧➭ ω ✲s❡♠✐ ♠ë s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭ r❡❣✉❧❛r ❣❡♥✲ X − A ❧➭ t❐♣ ω ✲♥ö❛ ➤ã♥❣ ❝❤Ý♥❤ q✉② s✉② ré♥❣ ✈➭ ✈✐Õt ❧➭ ω ✲sr❣✲♠ë✳ ❚❐♣ A ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭s❡♠✐✲r❡❣✉❧❛r ❣❡♥❡r❛❧✲ ✐③❡❞ ❝❧♦s❡❞✮ ♥Õ✉ scl(A) ⊂ U ✱ ✈í✐ ♠ä✐ t❐♣ ♠ë ❝❤Ý♥❤ q✉② U ♠➭ A⊂U ✈➭ ✈✐Õt ❧➭ sr❣✲➤ã♥❣✳ ✷✳✶✳✺ ệ ề ỗ t ó t ỗ t❐♣ ω ❣s✲➤ã♥❣ ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭❛✮ ●✐➯ sö ❑❤✐ ➤ã t❐♣ ω ❣s✲➤ã♥❣❀ A ❧➭ t❐♣ ❝♦♥ ➤ã♥❣ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ )✳ X − A ❧➭ t❐♣ ❝♦♥ ♠ë ❝ñ❛ X ✱ s✉② r❛ X − A ❧➭ t❐♣ ω ✲♥ö❛ ♠ë ✈➭ A ❧➭ ω ✲♥ö❛ ➤ã♥❣✳ ❉♦ ➤ã t❛ ❝ã A = sclω (A)✳ ●✐➯ sö U A ⊂ U ✳ ❙✉② r❛ sclω (A) ⊂ U ✳ ❱❐② A ❧➭ t❐♣ ω ❣s✲➤ã♥❣✳ ✷✷ ❧➭ t❐♣ ♠ë ❜✃t ❦ú ♠➭ ✭❜✮ ●✐➯ sö A ❧➭ t❐♣ ω ❣s✲➤ã♥❣ ✈➭ U ì ỗ t í q ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❜✃t ❦ú ♠➭ A ⊂ U✳ A ❧➭ t❐♣ ω ❣s✲➤ã♥❣ t❛ s✉② r❛ sclω (A) ⊂ U ✳ A ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣✳ ✷✳✶✳✻ ét ỗ t t sở ỗ t sở t srở ệ ề ỗ t ó s rộ í q✉② ❧➭ t❐♣ ♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉②✳ A⊂X ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❜✃t ❦ú ♠➭ ①Ðt ✷✳✶✳✷ t❛ ❝ã ✷✳✶✳✽ ❧➭ ❧➭ t❐♣ ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✈➭ U A ⊂ U ✳ ❑❤✐ ➤ã t❛ ❝ã sclω (A) ⊂ U ✳ ◆❤ê ◆❤❐♥ scl(A) ⊂ sclω (A)✳ ❱❐② A ❧➭ t❐♣ sr❣✲➤ã♥❣✳ ➜Þ♥❤ ❧ý✳ ●✐➯ sư A ❧➭ t❐♣ ❝♦♥ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ )✳ ❑❤✐ ➤ã A ω ✲sr❣✲♠ë ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ F ⊂ sintω (A) ✈í✐ ♠ä✐ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② F ♠➭ F ⊂ A✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö A ❧➭ t❐♣ ω ✲♥ö❛ ♠ë s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✈➭ F ➤ã♥❣ ❝❤Ý♥❤ q✉② ❜✃t ❦ú ♠➭ s✉② ré♥❣ ❝❤Ý♥❤ q✉②✱ ➤ã t❛ ❝ã F ⊂ A✳ X −F ❑❤✐ ➤ã t❛ ❝ã ❧➭ t❐♣ X − A ❧➭ t❐♣ ω ✲♥ö❛ ➤ã♥❣ ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ✈➭ X − A ⊂ X − F✳ ❉♦ sclω (X − A) ⊂ X − F ✳ ❱× sclω (X − A) = X − sintω (A)✱ t❛ s✉② r❛ F ⊂ sintω (A)✳ ◆❣➢ỵ❝ ❧➵✐✱ ➤Ĩ ❝❤ø♥❣ ♠✐♥❤ ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣✳ ❑❤✐ ➤ã ❝ã X −U A ❧➭ t❐♣ ω ✲sr❣✲♠ë t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ X − A ❚❤❐t ✈❐②✱ ❣✐➯ sö U ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ♠➭ ❧➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② ♠➭ X − U ⊂ sintω (A)✳ X − A ⊂ U✳ X − U ⊂ A✳ ❉♦ ➤ã tõ ❣✐➯ t❤✐Õt t❛ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ X − sintω (A) ⊂ U ✳ ❱× t❤Õ t❛ ❝ã sclω (X − A) ⊂ U ✳ ❱❐② X − A ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ ✈➭ A ❧➭ t❐♣ ω ✲sr❣✲♠ë✳ A ω ✲sr❣✲➤ã♥❣ (X, τ )✳ ✷✳✶✳✾ ➜Þ♥❤ ❧ý✳ ●✐➯ sö ❑❤✐ ➤ã sclω (A) − A ❦❤➠♥❣ ❝❤ø❛ t ó í q rỗ ủ t❐♣ X✳ ✷✸ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư F ❧➭ t❐♣ ❝♦♥ ➤ã♥❣ ❝❤Ý♥❤ q✉② ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) s❛♦ ❝❤♦ F ⊂ sclω (A) − A✳ ❑❤✐ ➤ã F ⊂ X − A ✈➭ ✈× t❤Õ A ⊂ X − F ✳ ❱× A ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ ✈➭ X − F ✈➭ ❞♦ ➤ã t❛ ❝ã t➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉②✱ ♥➟♥ sclω (A) ⊂ X − F F ⊂ X − sclω (A)✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ F ⊂ (X − sclω (A)) ∩ sclω (A) = φ✳ ❱❐② F = φ✳ ✷✳✶✳✶✵ ❍Ö q✉➯✳ ◆Õ✉ A ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ )✱ t❤× sclω (A) − A ❧➭ t❐♣ ω ✲sr❣✲♠ë✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö F A ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ✈➭ ❧➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② s❛♦ ❝❤♦ t❛ ❝ã F ⊂ sclω (A) − A✳ ❑❤✐ ó ị ý F = ì t❤Õ F ⊂ sintω (sclω (A) − A)✳ ❉♦ ➤ã✱ ♥❤ê ➜Þ♥❤ ❧ý ✷✳✶✳✽ t❛ s✉② r❛ sclω (A) − A ❧➭ t❐♣ ω ✲sr❣✲♠ë✳ ✷✹ ❈➳❝ ➳♥❤ ①➵ s✉② ré♥❣ tr➟♥ t❐♣ ✷✳✷ ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✷✳✷✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❛✮ ➳♥❤ ①➵ f : (X, τ ) −→ (Y, σ) ➤➢ỵ❝ ❣ä✐ ❧➭ ω✲♥ư❛ ❧✐➟♥ ω ✲s❡♠✐ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥t✐♥✉♦✉s✮ ✈➭ ✈✐Õt t➽t ❧➭ ω ❣s✲❧✐➟♥ tụ ế tụ s rộ ỗ t ó ➳♥❤ ①➵ f ✭❜✮ F tr♦♥❣ (Y, σ) t❛ ❝ã f −1 (F ) ❧➭ t❐♣ ω ❣s✲➤ã♥❣ tr♦♥❣ (X, τ )❀ : (X, τ ) −→ (Y, σ) ➤➢ỵ❝ ❣ä✐ ❧➭ ω ✲♥ư❛ ❧✐➟♥ tơ❝ s✉② ré♥❣ ω ✲s❡♠✐ r❡❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ❝♦♥t✐♥✉♦✉s✮ ✈➭ ✈✐Õt t➽t ❧➭ ω ✲sr❣✲❧✐➟♥ ❝❤Ý♥❤ q tụ ế ỗ t ó F tr (Y, σ) t❛ ❝ã f −1 (F ) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ✷✳✷✳✷ ➜Þ♥❤ ❧ý✳ ●✐➯ sö f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵✳ ❑❤✐ ➤ã ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ✭❛✮ f ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝❀ ✭❜✮ ị ủ ỗ t tr ứ (Y, σ) ❧➭ t❐♣ ω ✲sr❣✲♠ë tr♦♥❣ (X, τ )✳ (a) ⇒ (b)✳ ●✐➯ sö G ❧➭ t❐♣ ♠ë ❜✃t ❦ú tr♦♥❣ (Y, σ)✳ ❑❤✐ ➤ã Y −G ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ (Y, σ)✳ ❉♦ ➤ã✱ tõ ❣✐➯ t❤✐Õt t❛ s✉② r❛ f −1 (Y −G) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ▼➭ f −1 (Y − G) = X − f −1 (G)✱ ❞♦ ➤ã X − f −1 (G) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❱× ✈❐② f −1 (G) ❧➭ t❐♣ ω ✲sr❣✲♠ë tr♦♥❣ (X, τ )✳ (b) ⇒ (a)✳ ●✐➯ sö F ♠ë tr♦♥❣ tr♦♥❣ (Y, σ)✳ (X, τ )✳ ▼➭ ❧➭ t❐♣ ➤ã♥❣ ❜✃t ❦ú tr♦♥❣ ❉♦ ➤ã✱ tõ ❣✐➯ t❤✐Õt t❛ s✉② r❛ (Y, σ)✳ ❑❤✐ ➤ã Y − F ❧➭ t❐♣ f −1 (Y − F ) ❧➭ t❐♣ ω ✲sr❣✲♠ë f −1 (Y − F ) = X − f −1 (F )✱ ❞♦ ➤ã X − f −1 (F ) ❧➭ t❐♣ ω ✲sr❣✲♠ë tr♦♥❣ (X, τ )✳ ❱× ✈❐② f −1 (F ) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❱❐② f ❧➭ ➳♥❤ ①➵ ✷✳✷✳✸ ω ✲sr❣✲❧✐➟♥ tơ❝✳ ➜Þ♥❤ ❧ý✳ ◆Õ✉ f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝ ✈➭ h : (Y, σ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ t❤× ho f : (X, τ ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư ❧✐➟♥ tơ❝ ♥➟♥ E ❧➭ t❐♣ ➤ã♥❣ ❜✃t ❦ú tr♦♥❣ h−1 (E) ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ (Y, σ)✳ ✷✺ ▲➵✐ ✈× (Z, δ)✳ f ❧➭ ❉♦ h ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝✱ ♥➟♥ f −1 (h−1 (E)) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ▼➭ (ho f )−1 (E) = f −1 (h−1 (E)) ♥➟♥ (ho f )−1 (E) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❉♦ ➤ã ho f ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲ ❧✐➟♥ tơ❝✳ ∗ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❛✮ ❑❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ➤➢ỵ❝ ❣ä✐ ❧➭ ω ✲T ❦❤➠♥❣ ❣✐❛♥ ✷✳✷✳✹ ♥Õ✉ ♠ä✐ t❐♣ ω ✲sr❣✲➤ã♥❣ ❧➭ t❐♣ ➤ã♥❣❀ ✭❜✮ ❑❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ➤➢ỵ❝ ❣ä✐ ❧➭ ω ✲Tsrg ❦❤➠♥❣ ❣✐❛♥ ♥Õ✉ ♠ä✐ t❐♣ ω ✲sr❣✲➤ã♥❣ ❧➭ t❐♣ ω ❣s✲➤ã♥❣✳ ✷✳✷✳✺ (X, τ ) ➜Þ♥❤ ❧ý✳ ●✐➯ sö ✈➭ (Z, δ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✱ (Y, σ) ❧➭ ω ✲T 1∗ ❦❤➠♥❣ ❣✐❛♥✳ ❑❤✐ ➤ã ♥Õ✉ f : (X, τ ) −→ (Y, σ) ✈➭ h : (Y, σ) −→ (Z, δ) ω ✲sr❣✲❧✐➟♥ ❧➭ ❝➳❝ ➳♥❤ ①➵ tơ❝ t❤× ho f : (X, τ ) −→ (Z, δ) ❝ò♥❣ ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tơ❝✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư sr❣✲❧✐➟♥ tơ❝ ♥➟♥ ❧➭ t❐♣ ➤ã♥❣ ❜✃t ❦ú tr♦♥❣ (Z, δ)✳ ❉♦ h ❧➭ ➳♥❤ ①➵ ω ✲ h−1 (F ) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (Y, σ)✳ ▲➵✐ ❞♦ (Y, σ) ❧➭ ω ✲T 1∗ ❦❤➠♥❣ ❣✐❛♥ ♥➟♥ tô❝ t❛ s✉② r❛ F −1 h (F ) ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ (Y, σ)✳ ❱× t❤Õ✱ ♥❤ê f ❧➭ ω ✲sr❣✲❧✐➟♥ f −1 (h−1 (F )) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❱❐② ho f ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝✳ ✷✳✷✳✻ ➜Þ♥❤ ❧ý✳ ●✐➯ sư (X, τ ) ❧➭ ω ✲T 1∗ ❦❤➠♥❣ ❣✐❛♥✱ f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵ ➤ã♥❣✱ ❑❤✐ ➤ã ω ✲sr❣✲❧✐➟♥ tô❝ ❧➟♥ ✈➭ h : (Y, σ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ❜✃t ❦ú✳ ho f : (X, τ ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ h ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư f ❧➭ ➳♥❤ ①➵ ➤ã♥❣ ✈➭ ω ✲sr❣✲❧✐➟♥ tô❝ ❧➟♥✱ ho f ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝ ✈➭ A ❧➭ t❐♣ ➤ã♥❣ ❜✃t ❦ú tr♦♥❣ (Z, δ)✳ ❑❤✐ ➤ã (ho f )−1 (A) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✱ ❞♦ ➤ã f −1 (h−1 (A)) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ▼➭ (X, τ ) ❧➭ ω ✲T 1∗ ❦❤➠♥❣ ❣✐❛♥ ♥➟♥ f −1 (h−1 (A)) ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ (X, τ )✳ ❱× ✈❐② f (f −1 (h−1 (A))) ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ (Y, σ)✱ ❦Ð♦ t❤❡♦ h−1 (A) ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ (Y, σ)✳ ❱❐② h ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝✳ ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư h ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝✳ ❑❤✐ ➤ã ♥❤ê ❣✐➯ t❤✐Õt f tơ❝ ✈➭ ➜Þ♥❤ ❧ý ✷✳✷✳✸ t❛ s✉② r❛ ho f ❧➭ ➳♥❤ ①➵ ✷✻ ω ✲sr❣✲❧✐➟♥ tô❝✳ ❧➭ ω ✲sr❣✲❧✐➟♥ ✷✳✷✳✼ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✷❪✮ ➳♥❤ ①➵ f : (X, τ ) −→ (Y, σ) ➤➢ỵ❝ ❣ä✐ ❧➭ ❧✐➟♥ tơ❝ A ❝ñ❛ (Y, σ) t❛ ❝ã f −1 (A) ♠➵♥❤ ✭str♦♥❣❧② ts ế ỗ t (X, ) t❐♣ ♠ë ✈➭ ➤ã♥❣ tr♦♥❣ ✷✳✷✳✽ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❛✮ ➳♥❤ ①➵ f : (X, τ ) −→ (Y, σ) ➤➢ỵ❝ ❣ä✐ ❧➭ ω✲♥ö❛ ❧✐➟♥ ω ✲s❡♠✐ r❡❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ str♦♥❣❧② ❝♦♥t✐♥✲ tô❝ ♠➵♥❤ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭ ✉♦✉s✮ ✈➭ ✈✐Õt tt sr tụ ế ỗ t ω ✲sr❣✲♠ë A tr♦♥❣ (Y, σ) t❛ ❝ã f −1 (A) ❧➭ t❐♣ ♠ë tr♦♥❣ (X, τ )❀ ✭❜✮ ➳♥❤ ①➵ f : (X, τ ) −→ (Y, σ) ➤➢ỵ❝ ❣ä✐ ❧➭ ω ✲♥ư❛ ❤♦➭♥ t♦➭♥ ❧✐➟♥ tơ❝ ω ✲s❡♠✐ r❡❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ♣❡r❢❡❝t❧② ❝♦♥t✐♥✉♦✉s✮ ✈➭ ✈✐Õt s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭ t➽t ❧➭ ❝ã ω ✲sr❣✲❤♦➭♥ t♦➭♥ ❧✐➟♥ tô❝ ♥Õ✉ ỗ t srở A tr (Y, ) t f −1 (A) ❧➭ t❐♣ ♠ë ✈➭ ➤ã♥❣ tr♦♥❣ (X, τ )✳ ✷✳✷✳✾ ➜Þ♥❤ ❧ý✳ ✭❛✮ ◆Õ✉ f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tơ❝ ♠➵♥❤ t❤× ♥ã ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝❀ ✭❜✮ ◆Õ✉ f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ ♠➵♥❤ t❤× ♥ã ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tơ❝ ♠➵♥❤✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭❛✮ ●✐➯ sư ◆❤❐♥ ①Ðt ✷✳✶✳✻ t❛ ❝ã tơ❝ ♠➵♥❤✱ ♥➟♥ ✭❜✮ ●✐➯ sư (Y, σ)✳ ❑❤✐ ➤ã G ❧➭ t❐♣ ❝♦♥ ♠ë ❜✃t ❦ú tr♦♥❣ (Y, σ)✳ ❑❤✐ ➤ã ♥❤ê G ❧➭ t❐♣ ω ✲sr❣✲♠ë tr♦♥❣ (Y, σ)✳ ❱× f f −1 (G) ❧➭ t❐♣ ♠ë tr♦♥❣ (X, τ )✳ ❱❐② f f ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝ ♠➵♥❤ ✈➭ ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝✳ G ❧➭ t❐♣ ω ✲sr❣✲♠ë ❜✃t ❦ú tr♦♥❣ f −1 (G) ❧➭ t❐♣ ♠ë tr♦♥❣ (X, τ )✱ s✉② r❛ f ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tơ❝ ♠➵♥❤✳ ✷✳✷✳✶✵ ➜Þ♥❤ ❧ý✳ ➳♥❤ ①➵ f ỉ ế ỗ t t ó tr ω ✲sr❣✲➤ã♥❣ A tr♦♥❣ (Y, σ) t❛ ❝ã f −1 (A) ❧➭ t❐♣ (X, τ )✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö tr♦♥❣ : (X, τ ) −→ (Y, σ) ❧➭ ω ✲sr❣✲❧✐➟♥ tô❝ ♠➵♥❤ ♥Õ✉ f ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝ ♠➵♥❤✱ F (Y, σ)✳ ❑❤✐ ➤ã Y − F ❧✐➟♥ tô❝ ♠➵♥❤ ♥➟♥ ❧➭ t❐♣ ♠ë tr♦♥❣ ❧➭ t❐♣ ❧➭ t❐♣ ω ✲sr❣✲♠ë tr♦♥❣ (Y, σ)✳ ❱× f ω ✲sr❣✲➤ã♥❣ ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲ f −1 (Y − F ) ❧➭ t❐♣ ♠ë tr♦♥❣ (X, τ )✱ s✉② r❛ X − f −1 (F ) (X, τ )✳ ❱❐② f −1 (F ) ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ (X, τ )✳ ✷✼ ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (Y, σ)✳ ❚õ ❣✐➯ t❤✐Õt s✉② r❛ f −1 (Y − G) = X − f −1 (G) ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ ①➵ G ❧➭ t❐♣ ω ✲sr❣✲♠ë ❜✃t ❦ú tr♦♥❣ (Y, σ)✳ ❑❤✐ ➤ã Y − G ❧➭ (X, τ )✳ ❉♦ ➤ã f −1 (G) ❧➭ t❐♣ ♠ë tr♦♥❣ (X, τ )✳ ❱❐② f ❧➭ ➳♥❤ ω ✲sr❣✲❧✐➟♥ tơ❝ ♠➵♥❤✳ ✷✳✷✳✶✶ ➜Þ♥❤ ❧ý✳ ✭❛✮ ◆Õ✉ ♠➵♥❤ ✈➭ f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝ h : (Y, σ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tơ❝ t❤× ho f : (X, τ ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝❀ ✭❜✮ ◆Õ✉ ❧➭ ➳♥❤ ①➵ f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❤♦➭♥ t♦➭♥ ❧✐➟♥ tơ❝ t❤× ♥ã ω ✲sr❣✲❧✐➟♥ tơ❝ ♠➵♥❤✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭❛✮ ●✐➯ sö G ❧➭ t❐♣ ❝♦♥ ♠ë ❜✃t ❦ú ❝đ❛ (Z, δ)✳ ❱× h ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝ ♥➟♥ h−1 (G) ❧➭ t❐♣ ω ✲sr❣✲♠ë tr♦♥❣ (Y, σ)✳ ▲➵✐ ✈× f sr❣✲❧✐➟♥ tơ❝ ♠➵♥❤ t❛ ❝ã ❦ú tr♦♥❣ f ❧➭ ➳♥❤ ①➵ (Y, σ)✳ ω✲ f −1 (h−1 (G)) ♠ë tr♦♥❣ (X, τ )✱ s✉② r❛ (ho f )−1 (G) = f −1 (h−1 (G)) ❧➭ t❐♣ ♠ë tr♦♥❣ (X, τ )✳ ❱❐② ho f ✭❜✮ ●✐➯ sö ❧➭ ➳♥❤ ①➵ ❑❤✐ ➤ã ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝✳ ω ✲sr❣✲❤♦➭♥ t♦➭♥ ❧✐➟♥ tô❝ ✈➭ G ❧➭ t❐♣ ω ✲sr❣✲♠ë ❜✃t f −1 (G) ❧➭ t❐♣ ♠ë tr♦♥❣ (X, τ )✱ s✉② r❛ f ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tơ❝ ♠➵♥❤✳ ✷✳✷✳✶✷ ➜Þ♥❤ ❧ý✳ ●✐➯ sư f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵✳ ❑❤✐ ➤ã ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ✭❛✮ f ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❤♦➭♥ t♦➭♥ ❧✐➟♥ tô❝❀ ✭❜✮ ị ủ ỗ t tr Y F f (Y − F ) ➤ã♥❣ tr♦♥❣ tr♦♥❣ (Y, σ) ❧➭ t❐♣ ♠ë ✈➭ ➤ã♥❣ (X, τ )✳ ❈❤ø♥❣ ♠✐♥❤✳ ❑❤✐ ➤ã ω ✲sr❣✲➤ã♥❣ (a) ⇒ (b)✳ ●✐➯ sö F ω ✲sr❣✲➤ã♥❣ ❜✃t ❦ú tr♦♥❣ (Y, σ)✳ tr♦♥❣ (Y, σ)✳ ❉♦ ➤ã tõ ❣✐➯ t❤✐Õt t❛ s✉② r❛ ❧➭ t❐♣ ♠ë ✈➭ ➤ã♥❣ tr♦♥❣ (X, τ )✳ ❙✉② r❛ ❧➭ t❐♣ ω ✲sr❣✲♠ë ❧➭ t❐♣ f −1 (F ) ❧➭ t❐♣ ♠ë ✈➭ (X, τ )✳ (b) ⇒ (a)✳ ●✐➯ sö G ❧➭ t❐♣ ❝♦♥ ω ✲sr❣✲♠ë ❜✃t ❦ú tr♦♥❣ (Y, σ)✳ ❑❤✐ ➤ã Y −G ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (Y, σ)✳ ❉♦ ➤ã tõ ❣✐➯ t❤✐Õt t❛ s✉② r❛ f −1 (Y − G) ❧➭ t❐♣ ✷✽ ♠ë ✈➭ ➤ã♥❣ tr♦♥❣ ❱❐② f ❧➭ ➳♥❤ ①➵ ✷✳✷✳✶✸ ω ✲sr❣✲❤♦➭♥ t tụ ị ĩ ợ tt (X, τ )✱ ❦Ð♦ t❤❡♦ f −1 (G) ❧➭ t❐♣ ♠ë ✈➭ ➤ã♥❣ tr♦♥❣ (X, τ )✳ ➳♥❤ ①➵ f : (X, τ ) −→ (Y, σ) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ✭ω ✲s❡♠✐ ❣❡♥❡r❛❧✐③❡❞ ❝❧♦s❡❞ rrst ết s ợ ế ỗ t❐♣ ω s❣✲➤ã♥❣ F tr♦♥❣ (Y, σ) t❛ ❝ã f −1 (F ) ❧➭ t❐♣ ω s❣✲➤ã♥❣ tr♦♥❣ (X, τ )❀ ✭❜✮ ➳♥❤ ①➵ f : (X, τ ) −→ (Y, σ) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ ω✲♥ư❛ ➤ã♥❣ ω ✲s❡♠✐ r❡❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ❝❧♦s❡❞ ✐rr❡s♦❧✉t❡✮ ✈➭ ✈✐Õt t➽t s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭ ❧➭ ω ✲sr❣❝✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ ♥Õ✉ ỗ t sró F tr (Y, ) t ❝ã f −1 (F ) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, ) ị ý ế ợ tì ♥ã ❧➭ ➳♥❤ ①➵ ✭❜✮ ◆Õ✉ ❧➭ ➳♥❤ ①➵ f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵ ω ✲s❣❝✲❦❤➠♥❣ ❣✐➯✐ ω ✲sr❣✲❧✐➟♥ tô❝❀ f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣❝✲❦❤➠♥❣ ❣✐➯✐ ợ tì ó sr tụ ế f : (X, τ ) −→ (Y, σ) ✈➭ h : (Y, σ) −→ (Z, δ) ❧➭ ❝➳❝ ➳♥❤ ①➵ ω ✲sr❣❝✲❦❤➠♥❣ ợ tì ho f : (X, ) (Z, δ) ❝ò♥❣ ❧➭ ➳♥❤ ①➵ ω ✲sr❣❝✲ ❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭❛✮ ●✐➯ sư ✷✳✶✳✺ t❛ ❝ã F ❧➭ t❐♣ ➤➢ỵ❝ ♥➟♥ t❛ ❝ã F ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ ω s❣✲➤ã♥❣ tr♦♥❣ (Y, σ)✳ tr♦♥❣ f f ❧➭ ➳♥❤ ①➵ f −1 (F ) ❧➭ t❐♣ ω s❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❱❐② f ✭❜✮ ●✐➯ sư ❱× (Y, σ)✳ ❑❤✐ ➤ã ♥❤ê ▼Ư♥❤ ➤Ị ❧➭ ➳♥❤ ①➵ (Y, σ)✳ ❑❤✐ ➤ã F ❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ ♥➟♥ ❧➭ ➳♥❤ ①➵ ω ✲sr❣❝✲❦❤➠♥❣ ❧➭ t❐♣ ω ✲s❣❝✲❦❤➠♥❣ ❣✐➯✐ ❉♦ ➤ã f −1 (F ) ❧➭ t❐♣ ω ✲sr❣✲❧✐➟♥ tơ❝✳ ❣✐➯✐ ➤➢ỵ❝ ✈➭ F ❧➭ t❐♣ ➤ã♥❣ ❜✃t ❦ú ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (Y, σ)✳ ❱× f ❧➭ ➳♥❤ ①➵ f −1 (F ) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❱❐② f ω ✲sr❣❝✲ ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝✳ ✭❝✮ ❙✉② trù❝ t✐Õ♣ tõ ➤Þ♥❤ ♥❣❤Ü❛✳ f : (X, τ ) −→ (Y, ) s ị ý ế ợ h : (Y, σ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ω ❣s✲❧✐➟♥ tơ❝✱ t❤× ho f : (X, τ ) −→ ✷✾ ❧➭ ➳♥❤ ①➵ ❣✐➯✐ (Z, δ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tơ❝✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư F ❧➭ t❐♣ ➤ã♥❣ ❜✃t ❦ú tr♦♥❣ (Z, δ)✳ ω ❣s✲❧✐➟♥ tô❝ ♥➟♥ h−1 (F ) ❧➭ t❐♣ ω s❣✲➤ã♥❣ tr♦♥❣ (Y, σ)✳ s❣❝✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✱ ♥➟♥ ♥❤ê ▼Ư♥❤ ➤Ị ✷✳✶✳✺ t❛ ❝ã (X, τ )✳ ❱❐② ho f ✷✳✷✳✶✻ ❧➭ ➳♥❤ ①➵ f ❧➭ ➳♥❤ ①➵ ω✲ f −1 (h−1 (F )) ❧➭ t❐♣ ω s❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❑❤✐ ➤ã ω ✲sr❣✲❧✐➟♥ tô❝✳ (X, τ ) ✈➭ (Z, δ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✱ (Y, σ) ❧➭ ❦❤➠♥❣ ❣✐❛♥✳ ó ế ợ ì ➳♥❤ ①➵ (ho f )−1 (F ) = f −1 (h−1 (F )) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ ➜Þ♥❤ ❧ý✳ ●✐➯ sư ω ✲Tsrg h ❱× f : (X, τ ) −→ (Y, σ) h : (Y, σ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ ω ✲s❣❝✲ tơ❝✱ t❤× ho f : (X, τ ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö E ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (Y, σ)✳ ❧➭ t❐♣ ω s❣✲➤ã♥❣ tr♦♥❣ ❑❤✐ ➤ã h−1 (E) ❦❤➠♥❣ ❣✐❛♥✱ ♥➟♥ h−1 (E) ❧➭ t❐♣ ➤ã♥❣ ❜✃t ❦ú tr♦♥❣ (Y, σ)✳ ❉♦ ❱× (Y, σ) ❧➭ ω ✲Tsrg f ❧➭ ➳♥❤ ①➵ (Z, δ)✳ ω ✲s❣❝✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✱ t❛ ❝ã f −1 (h−1 (E)) ❧➭ t❐♣ ω ❣s✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❙✉② r❛ (ho f )−1 (E) = f −1 (h−1 (E)) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❱❐② ho f ✷✳✷✳✶✼ (Y, σ) ➜Þ♥❤ ❧ý✳ ●✐➯ sö ❧➭ ω ✲T 1∗ (X, τ )✱ (Y, σ) ❧➭ ➳♥❤ ①➵ ✈➭ ❦❤➠♥❣ ❣✐❛♥✳ ❑❤✐ ➤ã ♥Õ✉ ω ✲sr❣✲❧✐➟♥ tô❝✳ (Z, δ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✱ f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵ ω ✲s❣❝✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ ✈➭ h : (Y, σ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝✱ t❤× ho f : (X, τ ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tơ❝✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư ❧➭ t❐♣ ❧➭ ➳♥❤ ①➵ (X, τ )✳ ❱❐② (Z, δ)✳ ❑❤✐ ➤ã h−1 (F ) (Y, σ)✳ ❱× ✈❐② −1 h (F ) ❧➭ t❐♣ ω s❣✲➤ã♥❣ tr♦♥❣ (Y, ) ì s ợ f −1 (h−1 (F )) ❧➭ t❐♣ ω ❣s✲➤ã♥❣ tr♦♥❣ ❱× ✈❐② ho f ✷✳✷✳✶✽ ❧➭ t❐♣ ➤ã♥❣ ❜✃t ❦ú tr♦♥❣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (Y, σ)✳ ❉♦ (Y, σ) ❧➭ ω ✲T 1∗ ❦❤➠♥❣ ❣✐❛♥ ♥➟♥ h−1 (F ) ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ f F (ho f )−1 (F ) = f −1 (h−1 (F )) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❧➭ ➳♥❤ ①➵ ➜Þ♥❤ ❧ý✳ ω ✲sr❣✲❧✐➟♥ tụ f ế ỉ ế ỗ t❐♣ : (X, τ ) −→ (Y, σ) ❧➭ ω ✲sr❣❝✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ ω ✲sr❣✲♠ë U ω ✲sr❣✲♠ë tr♦♥❣ (X, τ )✳ ✸✵ tr♦♥❣ (Y, σ) t❛ ❝ã f −1 (U ) ❧➭ t❐♣ ❈❤ø♥❣ ♠✐♥❤✳ ❈➬♥✳ ●✐➯ sö ➤ã Y −U ➤➢ỵ❝ ♥➟♥ ❧➭ t❐♣ U ❧➭ t❐♣ ω ✲sr❣✲♠ë ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (Y, σ)✳ ❱× f ❜✃t ❦ú tr♦♥❣ ❧➭ ➳♥❤ ①➵ (Y, σ)✳ ❑❤✐ ω ✲sr❣❝✲❦❤➠♥❣ ❣✐➯✐ f −1 (Y − U ) = X − f −1 (U ) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❉♦ ➤ã f −1 (U ) ❧➭ t❐♣ ω ✲sr❣✲♠ë tr♦♥❣ (X, τ )✳ ➜đ✳ ●✐➯ sư ω ✲sr❣✲♠ë tr♦♥❣ tr♦♥❣ ❧➭ t❐♣ (Y, σ)✳ ω ✲sr❣✲➤ã♥❣ ❜✃t ❦ú tr♦♥❣ (Y, σ)✳ ❑❤✐ ➤ã Y − F ❚õ ❣✐➯ t❤✐Õt t❛ s✉② r❛ f −1 (Y − F ) ❧➭ t❐♣ ❧➭ t❐♣ ω ✲sr❣✲♠ë (X, τ )✳ ▼➭ f −1 (Y − F ) = X − f −1 (F ) ♥➟♥ X − f −1 (F ) ❧➭ t❐♣ ω ✲sr❣✲ ♠ë tr♦♥❣ ①➵ F (X, τ )✳ ❉♦ ➤ã f −1 (F ) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❱❐② f ❧➭ ➳♥❤ ω sr ợ ị ý sử (X, ) ✈➭ (Z, δ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✳ ❑❤✐ ➤ã ♥Õ✉ f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣❝✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ ✈➭ h : (Y, σ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tơ❝✱ t❤× ho f : (X, τ ) −→ (Z, δ) ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲ ❧✐➟♥ tô❝✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư ω ✲sr❣✲❧✐➟♥ tơ❝ t❛ ❝ã F h−1 (F ) ❧➭ t❐♣ ➤ã♥❣ ❜✃t ❦ú tr♦♥❣ ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (Z, δ)✳ (Y, σ)✳ ❱× h ❧➭ ➳♥❤ ①➵ ❉♦ f ❧➭ ➳♥❤ ①➵ ω ✲sr❣❝✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ ♥➟♥ (ho f )−1 (F ) = f −1 (h−1 (F )) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣ (X, τ )✳ ❱❐② ho f ❧➭ ➳♥❤ ①➵ ω ✲sr❣✲❧✐➟♥ tô❝✳ ✸✶ ❦Õt ❧✉❐♥ ❙❛✉ t❤ê✐ ❣✐❛♥ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ị✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ✈➭ ♥❣❤✐➟♠ ❦❤➽❝ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉ ✶✳ ❍Ö t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ö♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ✈Ị t❐♣ r❣ ω ✲➤ã♥❣✱ ❧í♣ ❝➳❝ ❤➭♠ r❣ω ✲❧✐➟♥ tơ❝✱ ❤➭♠ r❣ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✳ ✷✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ♠Ư♥❤ ➤Ị✱ tÝ♥❤ ❝❤✃t ✈➭ ➤Þ♥❤ ❧ý ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❝❤➢❛ ❝❤ø♥❣ ♠✐♥❤ ❤♦➷❝ ❝❤ø♥❣ ♠✐♥❤ ò s ợ tể ệ ị ý ị ❧ý ✶✳✶✳✷✵✱ ➜Þ♥❤ ❧ý ✶✳✶✳✷✶✱ ❇ỉ ➤Ị ✶✳✶✳✷✹✱ ➜Þ♥❤ ❧ý ✶✳✶✳✷✺✱ ❍Ư q✉➯ ✶✳✶✳✷✻✱ ➜Þ♥❤ ❧ý ✶✳✸✳✼✱ ➜Þ♥❤ ❧ý ✶✳✸✳✶✼✳ ✸✳ ➜➢❛ r❛ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ❦Õt q✉➯ ♠í✐ ✈Ị ❝➳❝ t❐♣ ω ✲♥ư❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ❞ù❛ tr➟♥ ❝➡ së ❝➳❝ ♥❤❐♥ ①Ðt✱ ❧➢✉ ý ➤➲ ❝ã tõ ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✳ ➜✐Ị✉ ♥➭② t❤Ĩ ❤✐Ư♥ ë ❝❤➢➡♥❣ ✷ ❝đ❛ ❧✉❐♥ ✈➝♥✳ ✸✷ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❏✳ ▲✳ ❑❡❧❧② ✭✶✾✼✸✮✱ ❚➠♣➠ ➤➵✐ ❝➢➡♥❣✱ ◆❤➭ ①✉✃t ❜➯♥ ➜❍ ✈➭ ❚❍❈◆✱ ❍➭ ◆é✐✳ ❬✷❪ ◆❣✉②Ô♥ ❚❤Þ ❚❤✉ ✭✷✵✵✽✮✱ ❱Ị ❝➳❝ t❐♣ ω ✲♥ư❛ ➤ã♥❣ s✉② ré♥❣ ✈➭ ❝➳❝ t❐♣ ω ✲➤ã♥❣ s✉② ré♥❣✱ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sÜ ❚♦➳♥ ❤ä❝✱ ➜➵✐ ❤ä❝ ❱✐♥❤✳ ❬✸❪ ❚✳ ❆✳ ❆❧✲❍❛✇❛r② ✭✷✵✵✹✮✱ ω ✲❣❡♥❡r❛❧✐③❡❞ ❝❧♦s❡❞ s❡ts✱ ■♥t❡r✳ ❏✳ ❆♣♣❧✳ ▼❛t❤✳✱ ✶✻ ✭✸✮✱ ✸✹✶✲✸✺✸✳ ❬✹❪ ❑✳ ❨✳ ❆❧✲❩♦✉❜✐ ✭✷✵✵✺✮✱ ❖♥ ❣❡♥❡r❛❧✐③❡❞ ▼❛t❤✳ ❙❝✐✳✱ ω ✲❝❧♦s❡❞ s❡ts✱ ■♥t❡r✳ ❏✳ ▼❛t❤✳ ❛♥❞ ✶✸✱ ✷✵✶✶✲✷✵✷✶✳ ❬✺❪ ❑✳ ❨✳ ❆❧✲❩♦✉❜✐ ❛♥❞ ❇✳ ❆❧✲◆❛s❤❡❢ ✭✷✵✵✸✮✱ ❚❤❡ t♦♣♦❧♦❣② ♦❢ ❆❧♠❛♥❛s❛❤✱ ω ✲♦♣❡♥ s✉❜s❡ts✱ ✾✱ ✶✻✾✲✶✼✾✳ ❬✻❪ ❙✳ ❆❧✲●❤♦✉r ✭✶✾✾✾✮✱ ❈❡rt❡✐♥ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s r❡❧❛t❡❞ t♦ ♣❛r❛❝♦♠♣❛❝✲ ♥❡ss✱ P❤✳ ❉✳ ❚❤❡s✐s✱ ❯♥✐✈❡rs✐t② ♦❢ ❏♦r❞❛♥✱ ❆♠♠❛♥✳ ❬✼❪ ❈✳ ❲✳ ❇❛❦❡r ✭✶✾✾✻✮✱ ❖♥ ♣r❡s❡r✈✐♥❣ ❣✲❝❧♦s❡❞ s❡ts✱ ❑②✉♥❣♣♦♦❦ ▼❛t❤✳ ❏✳✱ ✸✻ ✭✶✮✱ ✶✾✺✲✶✾✾✳ ❬✽❪ ❘✳ ❊♥❣❡❧❦✐♥❣ ✭✶✾✽✾✮✱ ●❡♥❡r❛❧ ❚♦♣♦❧♦❣②✱ ❙✐❣♠❛ ❙❡r✐❡s ✐♥ P✉r❡ ▼❛t❤❡♠❛t✲ ✐❝s✱ ❍❡❧❞❡r♠❛♥♥✱ ❇❡r❧✐♥✱ ●❡r♠❛♥②✱ ✻✳ ❬✾❪ ❍✳ ❩✳ ❍❞❡✐❜ ✭✶✾✽✾✮✱ ω ✲❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✱ ❉✐r❛s❛t ❏♦✉r♥❛❧✱ ✶✻ ✭✷✮✱ ✶✸✻✲ ✶✺✸✳ ❬✶✵❪ ❍✳ ❩✳ ❍❞❡✐❜ ✭✶✾✽✷✮✱ ω ✲❝❧♦s❡❞ ♠❛♣♣✐♥❣✱ ❘❡✈✐t❛ ❈♦❧♦♠❜✐❛ ❞❡ ▼❛t❡♠❛t✐❝❛s✱ ✶✻ ✭✶✲✷✮✱ ✻✺✲✼✽✳ ❬✶✶❪ ❋✳ ❍✳ ❑❤❡❞r ❛♥❞ ❚✳ ◆♦✐r✐ ✭✶✾✽✻✮✱ ❖♥ ▼❛t❤✳✱ θ✲✐rr❡s♦❧✉t❡ ❢✉♥❝t✐♦♥s✱ ■♥❞✐❛♥ ❏✳ ✷✽ ✭✸✮✱ ✷✶✶✲✷✶✼✳ ❬✶✷❪ ◆✳ ▲❡✈✐♥ ✭✶✾✻✵✮✱ ❙tr♦♥❣ ❝♦♥t✐♥✉✐t② ✐♥ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s✱ ❆♠❡r✳ ▼❛t❤✳ ▼♦♥t❤❧②✱ ✻✼✱ ✷✻✾✲✷✼✺✳ ✸✸ ❬✶✸❪ ❆❤✳ ❆❧✳ ❖♠❛r✐ ❛♥❞ ▼✳ ❙✳ ◆♦♦r❛♥✐ ✭✷✵✵✼✮✱ ❘❡❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ω ✲❝❧♦s❡❞ s❡ts✱ ■♥t❡r✳ ❏✳ ▼❛t❤✳ ❛♥❞ ▼❛t❤✳ ❙❝✐✳✱ ❆rt✐❝❧❡ ■❉ ✶✻✷✾✷✱ ✶✶✲♣❛❣❡s✳ ❬✶✹❪ ◆✳ P❛❧❛♥✐❛♣♣❛♥ ❛♥❞ ❑✳ ❈✳ ❘❛♦ ✭✶✾✾✸✮✱ ❘❡❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ❝❧♦s❡❞ s❡ts✱ ❑②✉♥❣♣♦♦❦ ▼❛t❤✳ ❏✳✱ ✸✸ ✭✷✮✱ ✷✶✶✲✷✶✾✳ ❬✶✺❪ ❆✳ ❘❛♥✐ ❛♥❞ ❑✳ ❇❛❧❛❝❤❛♥❞r❛♥ ✭✶✾✾✼✮✱ ❖♥ r❡❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ❝♦♥t✐♥✉♦✉s ♠❛♣s ✐♥ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s✱ ❑②✉♥❣♣♦♦❦ ▼❛t❤✳ ❏✳✱ ✸✼✱ ✸✵✺✲✸✶✹✳ ❬✶✻❪ ❙✳ ❲✐❧❧❛r❞ ✭✶✾✼✵✮✱ ●❡♥❡r❛❧ ❚♦♣♦❧♦❣②✱ ❆❞❞✐s♦♥✲❲❡s❧❡②✱ ❘❡❛❞✐♥❣✱ ▼❛ss✱ ❯❙❆✳ ✸✹