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 ❧✉❐♥

✸✷

❚➭✐ ❧✐Ö✉ 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♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ✈➭ 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 ❦❤➳✐ ♥✐Ö♠ ✈➭ tÝ♥❤ ❝❤✃t ✈Ò ❝➳❝ t❐♣ s❣✲➤ã♥❣ ✈➭ s❣✲♠ë✳ ❈➳❝ t❐♣
s❣✲➤ã♥❣ ợ ứ ột rộ r tr ữ ❣➬♥ ➤➞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❣
tơ❝✮✱ ➳♥❤ ①➵

ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭r❣ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✮ ✳✳✳

❚r➟♥ ❝➡ së ❜➭✐ ❜➳♦ ❝đ❛ ❆✳ ❘❛♥✐ ✈➭ ❑✳ ❇❛❧❛❝❤❛♥❞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♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ❝➳❝ t❐♣ r❣
❤➭♠ r❣

ω ✲❧✐➟♥ tơ❝✱ ❤➭♠ r❣ω ✲❦❤➠♥❣ ợ

ớ ụ í tr ợ trì 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❛ ❦❤➳✐ ♥✐Ư♠ t❐♣

❝❤Ý♥❤ q✉② ✈➭ ❝➳❝ ➳♥❤ ①➵ s✉② ré♥❣ tr➟♥ t❐♣

ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣

ω ✲♥ö❛ ➤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♦♥❣ ❦❤♦❛✱ ➤➷❝ ❜✐Ưt ❧➭ ❝➳❝ t❤➬② ❝➠ ❣✐➳♦ tr♦♥❣ tỉ ●✐➯✐ 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 ♥❤✐Ò✉ ❝è ❣➽♥❣✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉
sãt✳ ❈❤ó♥❣ t➠✐ r✃t ♠♦♥❣ ♥❤❐♥ ợ ữ ý ế ó ó ủ t
✈➭ ❜➵♥ ➤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

ớ ỗ

A



ợ ọ ể ọ st ủ

U

ợ ọ

xU

tì

A ế

U A ế ợ

ó ✭ω ✲❝❧♦s❡❞✮

✭❬✶✵❪✮ ♥Õ✉ ♥ã ❝❤ø❛ t✃t ❝➯ ❝➳❝ ➤✐Ó♠ ❝➠

➤ä♥❣ ❝đ❛ ♥ã✳
P❤➬♥ ❜ï ❝đ❛ t❐♣

✶✳✶✳✷

ω ✲➤ã♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ω ✲♠ë ✭ω ✲♦♣❡♥✮✳

▼Ư♥❤ ➤Ị✳ ✭❬✹❪✮ ❚❐♣ ❝♦♥

A

❝đ❛ ❦❤➠♥❣ ❣✐❛♥ t

(X, )

ở ế ỉ ế ớ ỗ x ∈ A tå♥ 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 ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ 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✳ ✭❬✺❪✮ ●✐➯ sö

(X, τ )

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ ✈➭

X ✳ ❑❤✐ ➤ã

✭❛✮

(τω )ω = τω ❀

✭❜✮

(τA )ω = (τω )A ✳
✹

A

tr➟♥

A✳

❧➭ t❐♣ ❝♦♥


ω ✲❜❛♦ ➤ã♥❣ ✈➭ ω ✲♣❤➬♥ tr♦♥❣ ❝ñ❛ t❐♣ A ị ĩ t tự clA intA
ú ợ ý ệ ❧➬♥ ❧➢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❀

✭❡✮ ế

AB



tì

cl (A) cl (B)

ị ĩ

A ủ ❦❤➠♥❣ ❣✐❛♥ 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 ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ (X, τ ) ➤➢ỵ❝ ❣ä✐ ❧➭

A = int(clA)✳

P❤➬♥ ❜ï ❝đ❛ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② ✭r❡❣✉❧❛r
❝❧♦s❡❞✮✳ ▼ét ❝➳❝❤ t➢➡♥❣ ➤➢➡♥❣✱ t❐♣

✶✳✶✳✽

A ❧➭ ó í q ế A = cl(intA)

ét ỗ t ở í q ở ỗ t ó í q

ó



ị ĩ

A

ủ t

(X, )

ợ ❣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 ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ 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 ở



A ủ t (X, ) ợ ọ



ị ♥❣❤Ü❛✳ ✭❬✶✸❪✮ ❚❐♣ ❝♦♥

ω ✲❝✲➤ã♥❣ ✭ω ✲❝✲❝❧♦s❡❞✮ ♥Õ✉ tå♥ t➵✐ t❐♣ B ⊂ A s❛♦ ❝❤♦ A = clω (B)✳
❚õ ❝➳❝ ➤Þ♥❤ ♥❣❤Ü❛ ✈➭ tÝ♥❤ ❝❤✃t ➤➲ ❜✐Õt t❛ ❝ã tể ết 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 ó X


ó

ữ



ớ t ợ ❝❤♦ ❜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♥❣ ❝đ❛ (X, τ )✳ ❑❤✐ ➤ã clω (A) − A
X✳

❦❤➠♥❣ ❝❤ø❛ ❜✃t ỳ 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♥❣ ♠✐♥❤✳ ●✐➯ sư

❝❤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 ứ t ỳ t ó í q

rỗ ♥➭♦ ❝ñ❛

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♥❣ ♠✐♥❤✳ ●✐➯ sư

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♥❣ ❝đ❛ (X, τ )✳ ◆Õ✉ B ⊆ X

✶✳✶✳✷✶

➜Þ♥❤ ❧ý✳ ✭❬✶✸❪✮ ❈❤♦

s❛♦ ❝❤♦

A ⊆ B ⊆ clω (A)✱ t❤× B

◆Õ✉

ω ✲➤ã♥❣✳

❝ị♥❣ ❧➭ t❐♣ r❣

B ❧➭ t❐♣ ❝♦♥ ❝đ❛ (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

♥➟♥

❝ị♥❣ ❝❤ø❛

A ♥➟♥ clω (A) = clω (B) ⊆ U ✳ ❱❐② B ❧➭ t❐♣ r❣ω ó
P ò ứ t tự


t r

ị ý ế

A ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝đ❛ (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❐♣ ❝♦♥

❑❤✐ ➤ã tõ ➜Þ♥❤ ❧ý t s

ì t ị ý t s r❛


✶✳✶✳✷✸

Y

❇ỉ ➤Ị✳ ✭❬✹❪✮ ◆Õ✉

❧➭ t❐♣ ❝♦♥ ❝đ❛

✶✳✶✳✷✹

❧➭ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ♠ë ❝đ❛ ❦❤➠♥❣ ❣✐❛♥

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♥❣ ♠✐♥❤✳ ●✐➯ sư

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♥❣ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ X ✱ t❤× A ∩ B
❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

❝❤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✉②✳

✾

t❤×


◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư
➤ã♥❣ ❝❤Ý♥❤ q✉②

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 × ) ì


ị ý ế

AìB

t r❣

ω ✲♠ë ❝ñ❛ (X, τ ) ✈➭ B

ω ✲♠ë ❝ñ❛ (Y, σ)✳

❧➭ t❐♣ ❝♦♥ r❣

❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

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 ở

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❛
2 ✈➭ t❐♣ ♠ë V ❝❤ø❛ ✹ s❛♦ ❝❤♦ (U × V ) − (A × B)
♠ë✮ ❝đ❛

❧➭ t❐♣ ế ợ
t ở

U



ứ

(U ì V ) (A ì B) ❧➭ t❐♣ ❦❤➠♥❣ ➤Õ♠ ➤➢ỵ❝ ✈í✐ ♠ä✐

2 ✈➭ ♠ä✐ t❐♣ ♠ë V

❝❤ø❛ ✹✳

✶✵


ω ✲T 1

✶✳✷

❦❤➠♥❣ ❣✐❛♥ s✉② ré♥❣ ❝❤Ý♥❤ q✉②

2

✶✳✷✳✶

➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✸❪✮ ❑❤➠♥❣ ❣✐❛♥

ω ✲T 21

s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✈✐Õt ❣ä♥ ❧➭ r❣

➤Ị✉ ❧➭ t❐♣

✶✳✷✳✷
✭❛✮

X

(X, τ ) ➤➢ỵ❝ ❣ä✐ ❧➭ ω ✲T 12
♥Õ✉ ♠ä✐ t❐♣ r❣

❦❤➠♥❣ ❣✐❛♥

ω ✲➤ã♥❣ tr♦♥❣ (X, τ )

ω ó

ị ý ớ ỗ (X, ) ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣

ω ✲T 12 ❀

❧➭ r❣

✭❜✮ ỗ tử
ứ

xX

ở

(a) (b) sử x ∈ X ✳ ◆Õ✉ {①} ❦❤➠♥❣ ❧➭ t❐♣ ❝♦♥ ➤ã♥❣

❝❤Ý♥❤ q✉②✳ ❑❤✐ ➤ã

X − {x} ❦❤➠♥❣ ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ✈➭ ❞♦ ➤ã ❝❤Ø ❝ã X

t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❝❤ø❛

(X, τ ) ❧➭ r❣ω ✲T 12

❧➭ ➤ã♥❣ ❝❤Ý♥❤ q✉② ❤♦➷❝

❧➭

X − {x}✳ ❱× ✈❐② X − {x} ❧➭ t❐♣ r❣ω ✲➤ã♥❣✳ ▼➷t ❦❤➳❝

❦❤➠♥❣ ❣✐❛♥✱ s✉② r❛

X − {x} ❧➭ t❐♣ ω ✲➤ã♥❣✳ ❱❐② {①} ❧➭ t❐♣

ω ✲♠ë✳
(b) ⇒ (a)✳ ●✐➯ sư A ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝đ❛ (X, τ ) ✈➭ x ∈ clω (A)✳ ❚❛ sÏ

❝❤Ø r❛ r➺♥❣
◆Õ✉

x ∈ A✳

{①} ❧➭ ➤ã♥❣ ❝❤Ý♥❤ q✉② ✈➭ x ∈
/ A t❤× x ∈ clω (A)−A✳ ❙✉② r❛ clω (A)−A

❝❤ø❛ ột t ó í q rỗ
ý
ế

{} ề ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ➜Þ♥❤

x ∈ A✳

{①} ❧➭ t❐♣ ω ✲♠ë t❤× ✈× x ∈ clω (A) t❛ ❝ã t❐♣ ω ✲♠ë U = {x} t❤♦➯ ♠➲♥

U ∩ A = ∅✳ ❑Ð♦ t❤❡♦ x ∈ A✳
❱❐② ❝➯ ❤❛✐ tr➢ê♥❣ ❤ỵ♣ t❛ ➤Ò✉ ❝ã

x ∈ A✳ ❉♦ ➤ã A = clω (A)✳ ❱❐② A ❧➭ t❐♣

ω ✲➤ã♥❣✳
✶✳✷✳✸

➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✻❪✮ ❑❤➠♥❣ ❣✐❛♥ t (X, ) ợ ọ ế ợ

ị ế ỗ t ở rỗ ề ế ợ

ó

ị ý

(X, ) ế ợ ị

(X, ) T1 ♥Õ✉ ♠ä✐ t❐♣ r❣ω ✲➤ã♥❣ ➤Ò✉ ❧➭ t❐♣ ω ✲➤ã♥❣✳
❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö

❦❤➠♥❣ ♠ë✱ ❦Ð♦ t❤❡♦

x∈X

✈➭

{①} ❦❤➠♥❣ ➤ã♥❣✳

❑❤✐ ➤ã

A = X − {x}

A ❧➭ t❐♣ r❣ω ✲➤ã♥❣✱ ❞♦ ❝❤Ø ❝ã ♠ét t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❝❤ø❛
✶✶


A ❧➭ X ✳ ❉♦ ➤ã tõ ❣✐➯ t❤✐Õt s✉② r❛ A ❧➭ t❐♣ ω ✲➤ã♥❣✱ ❦Ð♦ t❤❡♦ {①} ❧➭ ω ✲♠ë✳ ❱×
✈❐② tõ ▼Ư♥❤ ➤Ị ✶✳✶✳✷ s✉② r❛ tå♥ t➵✐
➤➢ỵ❝✳ ❉♦ ➤ã

U

U ∈τ

s❛♦ ❝❤♦

x∈U

✈➭

U − {x} ➤Õ♠

x ∈ X ột

t ở rỗ ế ợ ứ

t



ị ĩ

f : (X, ) (Y, σ) ➤➢ỵ❝ ❣ä✐ ❧➭
f (F ) ⊆

✭❛✮ ①✃♣ ①Ø ➤ã♥❣ ✭❛♣♣r♦①✐♠❛t❡❧② ❝❧♦s❡❞✮ ✭❬✼❪✮ ✈✐Õt ❣ä♥ ❧➭ ❛✲➤ã♥❣ ♥Õ✉

intA ✈í✐ ♠ä✐ t❐♣ ❝♦♥ ➤ã♥❣ F

❝ñ❛

X

✈➭

A ❧➭ t❐♣ ❝♦♥ ❣✲♠ë ❝ñ❛ Y

♠➭

f (F ) ⊆ A❀
✭❜✮ ①✃♣ ①Ø ❧✐➟♥ tô❝ ✭❛♣♣r♦①✐♠❛t❡❧② ❝♦♥t✐♥✉♦✉s✮ ✭❬✼❪✮ ✈✐Õt ❣ä♥ ❧➭ ❛✲❧✐➟♥ tô❝ ♥Õ✉

clA ⊆ f −1 (V ) ✈í✐ ♠ä✐ t❐♣ ❝♦♥ ♠ë V
X

♠➭

✭❝✮ ①✃♣ ①Ø

❝ñ❛

Y

✈➭

A ❧➭ t❐♣ ❝♦♥ ❣✲➤ã♥❣ ❝ñ❛

A ⊆ f −1 (V )❀

ω ✲➤ã♥❣ ✭❛♣♣r♦①✐♠❛t❡❧② ω ✲❝❧♦s❡❞✮ ✭❬✶✸❪✮ ✈✐Õt ❣ä♥ ❧➭ ❛✲ω ✲➤ã♥❣ ♥Õ✉

f (F ) ⊆ intω (A) ✈í✐ ♠ä✐ t❐♣ ❝♦♥ ➤ã♥❣ ❝❤Ý♥❤ q✉② F
❝♦♥ r❣

✭❞✮ ①✃♣ ①Ø

ω ✲♠ë ❝ñ❛ Y

♠➭

X

✈➭

A t❐♣

f (F ) ⊆ A❀

ω ✲❧✐➟♥ tô❝ ✭❛♣♣r♦①✐♠❛t❡❧② ω ✲❝♦♥t✐♥✉♦✉s✮ ✭❬✶✸❪✮ ✈✐Õt ❣ä♥ ❧➭ ❛✲ω ✲❧✐➟♥

tô❝ ♥Õ✉

clω (A) ⊆ f −1 (V ) ✈í✐ ♠ä✐ t❐♣ ❝♦♥ ♠ë ❝❤Ý♥❤ q✉② V

A ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝đ❛ X
✶✳✷✳✻

❝đ❛

❱Ý ❞ơ✳ ✭❬✶✸❪✮ ❈❤♦

♠➭

❝ñ❛

Y

✈➭

A ⊆ f −1 (V )✳

X = {a, b, c, d}

✈í✐ t➠♣➠ ➤➢ỵ❝ ❝❤♦ ❜ë✐ ❤ä

τ =

{∅, X, {a}, {b}, {a, b}, {a, b, c}} ✈➭ f : (X, ) (X, ) ợ
ị ❜ë✐
❉♦

X

f (a) = a✱ f (b) = d✱ f (c) = b✱ f (d) = c✳

❤÷✉ ❤➵♥ ♥➟♥

A = {b, c}

τω

❧➭ t➠♣➠ rê✐ r➵❝ ✈➭

❧➭ t❐♣ ❣✲♠ë ✈➭

♥❤➢♥❣

f (F ) ⊆ intA✳

✶✳✷✳✼

❱Ý ❞ô✳ ✭❬✶✸❪✮ ❈❤♦

✈➭

F = {c, d}

f

❑❤✐ ➤ã

f

ω ✲➤ã♥❣✳

❧➭ ❛✲

❦❤➠♥❣ ❧➭ ➳♥❤ ①➵ ❛✲➤ã♥❣✱ ✈× t❐♣

❧➭ t❐♣ ➤ã♥❣ t❤♦➯ ♠➲♥

f (F ) ⊆ A✱

X = R ✈í✐ t➠♣➠ ➤➢ỵ❝ ❝❤♦ ❜ë✐ τ = {∅, X, R − Q}

f : (X, τ ) −→ (X, τ ) ❧➭ ❤➭♠ ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ❜ë✐ f (x) = 0 ✈í✐ ♠ä✐ x ∈ X ✳

❑❤✐ ➤ã

f

❧➭ ❛✲➤ã♥❣✳ ❱í✐

F

❧➭ t❐♣ ➤ã♥❣ ❜✃t ❦ú ❝ñ❛

✶✷

X✱

❝❤Ø ❝ã ♠ét t❐♣ ❣✲♠ë


f (F ) ❧➭ X ✳

❝❤ø❛

❑❤✐ ➤ã

f

ω ✲➤ã♥❣✱ ✈× t❐♣ A = Q ❧➭ r❣ω ✲

❦❤➠♥❣ ❧➭ ➳♥❤ ①➵ ❛✲

F = R ❧➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② t❤♦➯ ♠➲♥ f (F ) ⊆ A✱ ♥❤➢♥❣ f (F ) ⊆

♠ë ✈➭

intω (A) = ∅✳
✶✳✷✳✽

♠ä✐ ❦❤➠♥❣ ❣✐❛♥

Y

t❤× ➳♥❤ ①➵

❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

q✉② ❝đ❛

f

❧➭ ❛✲

❦❤➠♥❣ ❣✐❛♥ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✈í✐

f : (X, τ ) −→ (Y, σ) ❧➭ ❛✲ω ✲❧✐➟♥ tô❝✳

ω ✲T 12

❧➭ r❣

❦❤➠♥❣ ❣✐❛♥ ✈➭

V

❧➭ t❐♣ ❝♦♥ ♠ë ❝❤Ý♥❤

ω ✲❧✐➟♥ tô❝✳
f

❧➭ t❐♣ ❝♦♥ ❝đ❛

❧➭ ❛✲

ω ✲❧✐➟♥ tơ❝✱ A ❧➭ t❐♣ ❝♦♥ r ó rỗ ủ

X ớ t = {, Y, A}✱ ❧✃② f : (X, τ ) −→ (Y, σ) ❧➭

➳♥❤ ①➵ ➤å♥❣ ♥❤✃t✳ ❚õ ❣✐➯ t❤✐Õt
tr♦♥❣

ω ✲T 21

A ❧➭ t❐♣ ω ✲➤ã♥❣✱ ❞ã ➤ã A = clω (A)✳ ì cl (A) f 1 (V )

ợ ❣✐➯ sö

X ✈➭ Y

X

❧➭ r❣

Y ✱ A ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝ñ❛ X s❛♦ ❝❤♦ A ⊆ f −1 (V ) ì X r T 12




X

ị ❧ý✳ ❑❤➠♥❣ ❣✐❛♥

X

✈➭ ♠ë tr♦♥❣

Y

f

s❛♦ ❝❤♦

❧➭ ❛✲

ω ✲❧✐➟♥ tô❝ ✈➭ ❞♦ A ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣

A ⊆ f −1 (A)✱ s✉② r❛ clω (A) ⊆ f −1 (A) = A✳

❉♦ ➤ã

A ❧➭ t❐♣ ω ✲➤ã♥❣ tr♦♥❣ X ✳ ❱× ✈❐② X

✶✳✷✳✾

❇ỉ ➤Ị✳ ◆Õ✉ ❝➳❝ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ✈➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② ❝đ❛

trï♥❣ ♥❤❛✉✱ t❤× t✃t ❝➯ ❝➳❝ t❐♣ ❝♦♥ ❝đ❛
❝❤ó♥❣ ❧➭ r❣

❧➭ r❣

X

ω ✲T 21

❦❤➠♥❣ ❣✐❛♥✳

ω ✲➤ã♥❣

❧➭ r❣

X

✭✈➭ ❦Ð♦ t❤❡♦ t✃t ❝➯

ω ✲♠ë✮✳

❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö

A ❧➭ t❐♣ ❝♦♥ ♥➭♦ ➤ã ❝ñ❛ X

t❐♣ ♠ë ❝❤Ý♥❤ q✉②✳ ❑❤✐ ➤ã

U

s❛♦ ❝❤♦

❧➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉②✱ ❞♦ ➤ã

A⊆U

U

✈í✐

U

❧➭

❧➭ t❐♣ ➤ã♥❣ ✈➭

clω (A) ⊆ clω (U ) ⊆ clU = U ì A t r ó


ị ý ế ❝➳❝ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ✈➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② ❝đ❛

trï♥❣ ♥❤❛✉✱ t❤× ❤➭♠

❧➭ t❐♣

f : (X, τ ) −→ (Y, σ) ❧➭ ❛✲ω ✲➤ã♥❣ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ f (F )

ω ✲♠ë ✈í✐ ♠ä✐ t❐♣ ❝♦♥ ➤ã♥❣ ❝❤Ý♥❤ q✉② F

❈❤ø♥❣ ♠✐♥❤✳ ❈➬♥✳ ●✐➯ sư

t❐♣ ❝♦♥ ❝đ❛
❝ã

Y

Y

f

❧➭ ❛✲

❝đ❛

X✳

ω ✲➤ã♥❣✳ ◆❤ê ❇ỉ ➤Ị ✶✳✷✳✾ s✉② r❛ t✃t ❝➯ ❝➳❝

ω ✲♠ë✳ ❱× ✈❐② ✈í✐ t❐♣ ❝♦♥ ➤ã♥❣ ❝❤Ý♥❤ q✉② ❜✃t ❦ú F

❧➭ r❣

❝ñ❛

X t❛

f (F ) ❧➭ t❐♣ r❣ω ✲♠ë ❝ñ❛ Y ✳ ❉♦ f ❧➭ ❛✲ω ✲➤ã♥❣ ♥➟♥ t❛ ❝ã f (F ) ⊆ intω (f (F ))

❦Ð♦ t❤❡♦

f (F ) = intω (f (F ))✱ ❞♦ ➤ã f (F ) ❧➭ t❐♣ ω ✲♠ë✳

✶✸


➜đ✳ ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư
q✉② ✈➭

f (F ) ❧➭ t❐♣ ω ✲♠ë✱ f (F ) ⊆ A ✈í✐ ♠ä✐ F

➤ã♥❣ ❝❤Ý♥❤

A ❧➭ r❣ω ✲♠ë✳ ❑❤✐ ➤ã t❛ ❝ã f (F ) = intω (f (F )) ⊆ intω (A)✳ ❱❐② f

❧➭

ω ✲➤ã♥❣✳

❛✲

✶✳✷✳✶✶

➜Þ♥❤ ❧ý✳ ◆Õ✉ ❝➳❝ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ✈➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② ❝đ❛

trï♥❣ ♥❤❛✉✱ t❤× ❤➭♠

X

f : (X, τ ) −→ (Y, σ) ❧➭ ❛✲ω ✲❧✐➟♥ tô❝ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉

f −1 (V ) ❧➭ t❐♣ ω ✲➤ã♥❣ ✈í✐ ♠ä✐ t❐♣ ❝♦♥ ♠ë ❝❤Ý♥❤ q✉② V
❈❤ø♥❣ ♠✐♥❤✳ ❚➢➡♥❣ tù ➜Þ♥❤ ❧ý ✶✳✷✳✶✵✳

✶✹

❝đ❛

Y✳


❤➭♠ r❣

✶✳✸

✶✳✸✳✶
✭❛✮

➳♥❤ ①➵ f : (X, τ ) −→ (Y, ) ợ ọ

ị ĩ

tụ ế f −1 (V ) ❧➭ t❐♣ ω ✲♠ë tr♦♥❣ (X, τ ) ✈í✐ ♠ä✐ t❐♣ ♠ë V
❝đ❛

✭❜✮

ω ✲❧✐➟♥ tơ❝

(Y, σ)❀

ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ ✭❬✸❪✮ ♥Õ✉ f −1 (F ) ❧➭ t❐♣ ω ✲➤ã♥❣ tr♦♥❣ (X, τ ) ✈í✐ ♠ä✐
t❐♣

ω ✲➤ã♥❣ F

❝đ❛

(Y, σ)❀

ω ✲❧✐➟♥ tô❝ ✭❬✹❪✮ ♥Õ✉ f −1 (F ) ❧➭ t❐♣ ❣ω ✲➤ã♥❣ ❝đ❛ (X, τ ) ✈í✐ ♠ä✐ t❐♣ ➤ã♥❣

✭❝✮ ❣

F
✭❞✮ ❣

❝đ❛

(Y, σ)❀

ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ ✭❬✹❪✮ ♥Õ✉ f −1 (F ) ❧➭ t❐♣ ❣ω ✲➤ã♥❣ ❝ñ❛ (X, τ ) ✈í✐ ♠ä✐
t❐♣ ❣

ω ✲➤ã♥❣ F

❝đ❛

✭❡✮ ❘✲➳♥❤ ①➵ ✭❬✶✶❪✮ ♥Õ✉

(Y, σ)❀

f −1 (F ) ❧➭ t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉② tr♦♥❣ (X, τ ) ✈í✐ ♠ä✐

t❐♣ ➤ã♥❣ ❝❤Ý♥❤ q✉②

✶✳✸✳✷
✭❛✮ ❣

➜Þ♥❤ ♥❣❤Ü❛✳

F

❝đ❛

(Y, σ)✳

➳♥❤ ①➵ f : (X, τ ) −→ (Y, σ) ➤➢ỵ❝ ❣ä✐ ❧➭

ω ✲➤ã♥❣ ✭❬✹❪✮ ♥Õ✉ f (F ) ❧➭ t❐♣ ❣ω ✲➤ã♥❣ tr♦♥❣ (Y, σ) ✈í✐ ♠ä✐ t❐♣ ➤ã♥❣ F
❝ñ❛

✭❜✮ r❣

(X, τ )❀

ω ✲➤ã♥❣ ✭❬✶✸❪✮ ♥Õ✉ f (F ) ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ (Y, σ) ✈í✐ ♠ä✐ t❐♣ ➤ã♥❣

F

❝đ❛

(X, τ )❀

✭❝✮ r♦✲❜➯♦ t♦➭♥ ✭❬✶✸❪✮ ♥Õ✉
♠ë ❝❤Ý♥❤ q✉②

✭❞✮ t✐Ị♥

F

V

f (V ) ❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② tr♦♥❣ (Y, σ) ✈í✐ ♠ä✐ t❐♣

❝đ❛

(X, τ )❀

ω ✲➤ã♥❣ ✭❬✶✸❪✮ ♥Õ✉ f (F ) ❧➭ t❐♣ ω ✲➤ã♥❣ ❝đ❛ (Y, σ) ✈í✐ ♠ä✐ t❐♣ ω ✲➤ã♥❣
❝ñ❛

(X, τ )✳

✶✺


✶✳✸✳✸

❱Ý ❞ơ✳ ✭❬✶✸❪✮ ❈❤♦

X = {a, b, c, d}

✈í✐ t➠♣➠ ➤➢ỵ❝ ❝❤♦ ❜ë✐ ❤ä

τ =

{∅, X, {a}, {b}, {a, b}, {a, b, c}} ✈➭ f : (X, τ ) −→ (X, ) ợ
ị ở

f (a) = a✱ f (b) = b✱ f (c) = d✱ f (d) = c✳ ❑❤✐ ➤ã f

t❐♣ t✃t ❝➯ ❝➳❝ t❐♣ ở í q ủ
t ị ĩ

r t ì

X {∅, X, {a}, {b}}✳ ◆❤➢♥❣ ♥Õ✉ ❝❤ó♥❣

g : (X, τ ) −→ (X, τ ) ❝❤♦ ❜ë✐ g(a) = c✱ g(b) = d✱ g(c) = a✱

g(d) = b✱ t❤× g ❦❤➠♥❣ ❧➭ ➳♥❤ ①➵ r♦✲❜➯♦ t♦➭♥✳
✶✳✸✳✹
tơ❝ ✭r❣

t❐♣

➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✸❪✮

ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✮ ♥Õ✉ f −1 (F ) ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ (X, τ ) ✈í✐ ♠ä✐

ω ✲➤ã♥❣ ✭r❣ω ✲➤ã♥❣✮ F

✶✳✸✳✺

➳♥❤ ①➵ f : (X, τ ) −→ (Y, σ) ợ ọ r
ủ

(Y, )

ét ỗ tụ

tụ ỗ

tụ tụ
ỗ

ợ tụ

ỗ

ợ tụ

ỗ r

ị ý

ợ r tụ

(X, τ ) ❧➭ r❣ω ✲T 21

❦❤➠♥❣ ❣✐❛♥ ✈➭ ➳♥❤ ①➵

f : (X, τ ) −→

(Y, σ)✳ ❑❤✐ ➤ã
✭❛✮ ◆Õ✉

f

❧➭ ❣

ω ✲❧✐➟♥ tơ❝ t❤× f

✭❜✮ ◆Õ✉

f

❧➭ r❣

✭❝✮ ◆Õ✉

f

❧➭ r❣

✭❞✮ ◆Õ✉

f

❧➭ ❣

✭❡✮ ◆Õ✉

f

❧➭ r❣

ω ✲❧✐➟♥ tơ❝❀

ω ✲❧✐➟♥ tơ❝ t❤× f

❧➭

ω ✲❧✐➟♥ tơ❝ tì f



ợ
tụ

ợ tì f

ợ tì f

ứ ●✐➯ sư

❣

❧➭

ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝❀

❧➭

❧➭

ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✳

A ❧➭ t❐♣ ❝♦♥ ➤ã♥❣ ❜✃t ❦ú ❝đ❛ Y ✳

❱×

f

❧➭ ➳♥❤ ①➵

ω ✲❧✐➟♥ tô❝ s✉② r❛ f −1 (A) ❧➭ t❐♣ ❝♦♥ ❣ω ✲➤ã♥❣ ❝đ❛ X ✳ ◆❤ê ➜Þ♥❤ ❧ý ✶✳✶✳✶✹ s✉②

r❛

f −1 (A) ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝ñ❛ X ✳

r❛

f −1 (A) ❧➭ t❐♣ ❝♦♥ ω ✲➤ã♥❣ ❝đ❛ X ✳ ❱× ✈❐② f
✶✻

❉♦

(X, τ ) ❧➭ r❣ω ✲T 12
❧➭ ➳♥❤ ①➵

❦❤➠♥❣ ❣✐❛♥ s✉②

ω ✲❧✐➟♥ tô❝✳


✭❜✮ ●✐➯ sö

A ❧➭ t❐♣ ❝♦♥ ω ✲➤ã♥❣ ❜✃t ❦ú ❝đ❛ Y ✳ ❱× f

s✉② r❛

f −1 (A) ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝ñ❛ X ✳

s✉② r❛

f −1 (A)

❧➭ t❐♣ ❝♦♥

ω ✲➤ã♥❣

❝đ❛

X✳

❉♦

ω ✲❧✐➟♥ tơ❝

❧➭ ➳♥❤ ①➵ r❣

(X, τ ) ❧➭ r T 12

ì

f









ợ

A t ❝♦♥ ➤ã♥❣ ❜✃t ❦ú ❝ñ❛ Y ✳ ❙✉② r❛ A ❧➭ t❐♣ ω ✲➤ã♥❣ tr♦♥❣

✭❝✮ ●✐➯ sư

Y ✳ ❱× f

❧➭ ➳♥❤ ①➵ r❣

(X, τ ) ❧➭ r❣ω ✲T 21

ω ✲❧✐➟♥ tô❝ s✉② r❛ f −1 (A) ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝ñ❛ X ✳ ❉♦
f −1 (A) ❧➭ t❐♣ ❝♦♥ ω ✲➤ã♥❣ ❝ñ❛ X ✳ ❙✉② r❛

❦❤➠♥❣ ❣✐❛♥ s✉② r❛

f −1 (A) ❧➭ t❐♣ ❣ω ✲➤ã♥❣ tr♦♥❣ X ✳ ❱× ✈❐② f

❧➭ ➳♥❤ ①➵ ❣

ω ✲❧✐➟♥ tô❝✳

✭❞✮ ✈➭ ✭❡✮ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ✭❝✮✳

✶✳✸✳✼

➜Þ♥❤ ❧ý✳ ❈❤♦

(Y, σ) ❧➭ r❣ω ✲T 12

❦❤➠♥❣ ❣✐❛♥ ✈➭ ➳♥❤ ①➵

f : (X, τ ) −→

(Y, ) ó
ế

f



ợ tì f

ế

f



ợ



ợ tì f

ợ

r

ứ tự ị ý



ị ❧ý✳ ❈❤♦

➤➢ỵ❝ ✈➭ t✐Ị♥

f : (X, τ ) −→ (Y, σ) ❧➭ ♠ét t♦➭♥ ➳♥❤✱ r❣ω ✲❦❤➠♥❣ ❣✐➯✐

ω ✲➤ã♥❣✳

◆Õ✉

X

❧➭ r❣

ω ✲T 12

❦❤➠♥❣ ❣✐❛♥✱ t❤×

Y

ω ✲T 21

❝ị♥❣ ❧➭ r❣

❦❤➠♥❣ ❣✐❛♥✳
❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ s✉② r❛
❦❤➠♥❣ ❣✐❛♥ s✉② r❛
✈➭ t✐Ị♥
r❣

ω ✲T 21

✶✳✸✳✾

f −1 (A) ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝ñ❛ X ✳

ω✲

❧➭ ➳♥❤ ①➵ r❣

❱×

X

f −1 (A) ❧➭ t❐♣ ❝♦♥ ω ✲➤ã♥❣ ❝ñ❛ X ✳ ▼➷t ❦❤➳❝ f

ω ✲T 12

❧➭ r❣

❧➭ t♦➭♥ ➳♥❤

ω ✲➤ã♥❣ ♥➟♥ f (f −1 (A)) = A ❧➭ t❐♣ ❝♦♥ ω ✲➤ã♥❣ ❝ñ❛ Y ✳ ❱❐② Y

ũ



ị ý

ớ ỗ


A t r ó ❝ñ❛ Y ✱ ❞♦ f

A ⊆ Y

ω ✲♠ë V

❝ñ❛

Y

➳♥❤ ①➵ f

: (X, ) (Y, )

ỗ t ở
s ❝❤♦

A⊆V

U

✈➭

❝❤ø❛

f −1 (A)

f −1 (V ) ⊆ U ✳

✶✼

ω ✲➤ã♥❣

❧➭ ❣

♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉

➤Ò✉ tå♥ t➵✐ ♠ét t❐♣ ❝♦♥


♠ë ❝❤ø❛
❝ñ❛

Y

✈➭

Y

❧➭ ♠ét ➳♥❤ ①➵ ❣

U

❧➭ ♠ét t❐♣

f −1 (V ) ⊆ U ✳

s❛♦ ❝❤♦

F

❧➭ ♠ét t❐♣ ❝♦♥ ➤ã♥❣ ❝ñ❛

X

✈➭

H

❧➭ ♠ét t❐♣ ❝♦♥ ♠ë

f (F ) ⊆ H ✳ ❑❤✐ ➤ã f −1 (Y − f (F )) ⊆ X − F

t❤❡♦ ❣✐➯ t❤✐Õt ♥➟♥ tå♥ t➵✐ ♠ét t❐♣ ❝♦♥ ❣
✈➭

✈➭

f −1 (A)✳ ❑❤✐ ➤ã V = Y − f (X − U ) ❝ò♥❣ ❧➭ t❐♣ ❝♦♥ ❣ω ✲♠ë ❝❤ø❛ A

◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư
❝đ❛

ω ✲➤ã♥❣✱ A ⊆ Y

f

❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö

f −1 (V ) ⊆ X − F ✳

❉♦ ➤ã

ω ✲♠ë V

❝ñ❛

Y

s❛♦ ❝❤♦

✈➭

X −F

♠ë

Y − f (F ) ⊆ V

F ⊆ X − f −1 (V ) s✉② r❛ f (F ) ⊆ Y − V ✳

❱×

Y −H ⊆ Y −f (F ) s✉② r❛ f −1 (Y −H) ⊆ f −1 (Y −f (F )) ⊆ f −1 (V ) ⊆ X−F ✱
❞♦ ➤ã

F ⊆ X − f −1 (V ) ⊆ X − f −1 (Y − f (F )) ⊆ X − f −1 (Y − H)✳ ❱× ✈❐②

f (F ) ⊆ Y −V ⊆ H ✳ ❉♦ Y −V
H

❦Ð♦ t❤❡♦

✶✳✸✳✶✵

t❐♣

ω ✲➤ã♥❣ ✈➭ clω (f (F )) ⊆ clω (Y −V ) ⊆

f (F ) ❧➭ t❐♣ ❣ω ✲➤ã♥❣✳ ❱❐② f

➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✸❪✮

➤ã♥❣ ✭r❣

❧➭ t❐♣ ❣

➳♥❤ ①➵ f

❧➭ ➳♥❤ ①➵ ❣

ω ✲➤ã♥❣✳

: (X, τ ) −→ (Y, σ) ➤➢ỵ❝ ❣ä✐ ❧➭ ❣ω ✲❝✲

ω ✲❝✲➤ã♥❣✮ ♥Õ✉ f (A) ❧➭ t❐♣ ❣ω ✲➤ã♥❣ ✭r❣ω ✲➤ã♥❣✮ tr♦♥❣ (Y, σ) ớ ọ

ó A ủ (X, )
ét ỗ ó ó ỗ ①➵ ❣ω ✲

✶✳✸✳✶✶

➤ã♥❣ ❧➭ r❣

✶✳✸✳✶✷
❑❤✐ ➤ã

ω ✲➤ã♥❣✳
f : (X, τ ) −→ (Y, σ) ❧➭ ❘✲➳♥❤ ①➵ ✈➭ r❣ω ✲❝✲➤ã♥❣✳

➜Þ♥❤ ❧ý✳ ●✐➯ sö

f (A) ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ Y

❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

❝❤Ý♥❤ q✉② ❝đ❛

Y

A ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❝đ❛ X
f (A) ⊆ U ✳ ❱× f

s❛♦ ❝❤♦

A ⊆ f −1 (U )✳

♠ë ❝❤Ý♥❤ q✉② ❝ñ❛

X

f −1 (U )✳

f (clω (A)) ⊆ U ✳

❑Ð♦ t❤❡♦

ω ✲❝✲➤ã♥❣

❧➭ ➳♥❤ ①➵ r❣

✈➭

✈í✐ ♠ä✐ t❐♣ ❝♦♥ r❣

♥➟♥

U

❧➭ t❐♣ ❝♦♥ ♠ë

f −1 (U ) ❧➭ t❐♣ ❝♦♥

A ❧➭ t❐♣ r❣ω ✲➤ã♥❣ ♥➟♥ clω (A) ⊆

▼➷t ❦❤➳❝

f (clω (A))

✈➭

❧➭ ❘✲➳♥❤ ①➵ ♥➟♥

❉♦

ω ✲➤ã♥❣ A ❝ñ❛ X ✳

clω (A)

ω ✲➤ã♥❣✳

❧➭ t❐♣ r❣

❧➭ t❐♣

ω ✲❝✲➤ã♥❣

❉♦ ➤ã

✈➭

f

clω (f (A)) ⊆

clω (f (clω (A))) ⊆ U ✳

❱❐②

✶✳✸✳✶✸
✈➭

f (A) ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ tr♦♥❣ Y ✳
➜Þ♥❤ ❧ý✳ ●✐➯ sư

ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝✳

f : (X, τ ) −→ (Y, σ)

◆Õ✉

B

❧➭ t❐♣ r❣

✶✽

❧➭ t♦➭♥ ➳♥❤✱ r♦✲❜➯♦ t♦➭♥

ω ✲➤ã♥❣ tr♦♥❣ Y ✱ t❤× f −1 (B) ❧➭ t❐♣


r❣

ω ✲➤ã♥❣ tr♦♥❣ X ✳
❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö

f −1 (B) ⊆ G✳
s✉② r❛

f (G)

❑❤✐ ➤ã

ω ✲➤ã♥❣

t❐♣

B ⊆ f (G)

✈➭

❉♦

X

❧➭ ♠ét t❐♣ ❝♦♥ ♠ë ❝❤Ý♥❤ q✉② ❝ñ❛

♠ë ❝❤Ý♥❤ q✉②✳

f −1 (clω (B)) ⊆ G✳

✈➭

G

f

❱×
❧➭

B

f

✈➭ ✈×

❧➭ t❐♣ r❣

ω ✲❦❤➠♥❣

s❛♦ ❝❤♦

❧➭ ➳♥❤ ①➵ r♦✲❜➯♦ t♦➭♥ ♥➟♥ t❛

ω ✲➤ã♥❣

♥➟♥

clω (B) ⊆ f (G)
f −1 (clω (B))

❣✐➯✐ ➤➢ỵ❝ s✉② r❛

clω (f −1 (clω (B))) = f −1 (clω (B))✳

❱× ✈❐②

❧➭

clω (f −1 (B)) ⊆

clω (f −1 (clω (B))) ⊆ G✳ ❉♦ ➤ã f −1 (B) ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ X ✳
✶✳✸✳✶✹

➜Þ♥❤ ❧ý✳ ●✐➯ sư

❦❤➠♥❣ ợ ế
ó tr

A

tr

Y

tì

ó

f 1 (A)

t r

A t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ Y

✈➭

f −1 (A) ⊆ U

✈í✐

U

❧➭

X ✳ ❑❤✐ ➤ã t❛ ❝ã X − U ⊆ X − f −1 (A) ⊆ f −1 (Y − A)

t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❝đ❛

f (X −U ) ⊆ Y −A✳ ❱× f ❧➭ ❛✲ω ✲➤ã♥❣ t❛ s✉② r❛ f (X −U ) ⊆ intω (Y −A) =

Y − clω (A)✳
❦❤➳❝

f

❧➭

❙✉② r❛

❱× ✈❐②

✶✳✸✳✶✺

X − U ⊆ X − f −1 (clω (A)) ✈➭ f −1 (clω (A)) ⊆ U ✳

✈➭

clω (f −1 (A)) ⊆ clω (f −1 (clω (A))) = f −1 (clω (A)) ⊆ U ✳

clω (f −1 (A)) ⊆ U
➜Þ♥❤ ❧ý✳ ◆Õ✉

t❐♣ ❝♦♥ ❣✲➤ã♥❣ ❝ñ❛

✈➭

f −1 (U )

clA ⊆ f −1 (U )

f −1 (A) ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ X ✳

f : (X, τ ) −→ (Y, σ) ❧➭ ❘✲➳♥❤ ①➵ ✈➭ r❣ω ✲➤ã♥❣✱ A ❧➭

X ✱ t❤× f (A) ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ Y ✳

❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

❑❤✐ ➤ã

▼➷t

ω ✲❦❤➠♥❣ ❣✐➯✐ ➤➢ỵ❝ ♥➟♥ f −1 (clω (A)) ❧➭ t❐♣ ω ✲➤ã♥❣✱ ❞♦ ➤ã f −1 (A) ⊆

f −1 (clω (A)) ⊆ U

r❣

ω ✲➤ã♥❣

❧➭ t❐♣ r❣

❧➭ ➳♥❤ ①➵ ❛✲

X✳

❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö

✈➭

f : (X, τ ) −→ (Y, σ)

f (A) ⊆ U

✈í✐

U

❧➭ t❐♣ ❝♦♥ ♠ë ❝❤Ý♥❤ q✉② ❝đ❛

❧➭ t❐♣ ❝♦♥ ♠ë ❝❤Ý♥❤ q✉② ❝❤ø❛
✈➭ ❞♦ ➤ã

f (clA) ⊆ U ✳

❉♦

f

A✳

❱×

A

❧➭ t❐♣ ❣✲➤ã♥❣ t❛ ❝ã

ω ✲➤ã♥❣

❧➭ r❣

Y✳

s✉② r❛

f (clA)

❧➭

ω ✲➤ã♥❣✳ ❱× t❤Õ t❛ ❝ã clω (f (clA)) ⊆ U ✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ clω (f (A)) ⊆ U ✳

❱❐②

f (A) ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ Y ✳

✶✳✸✳✶✻

➜Þ♥❤ ❧ý✳ ◆Õ✉

f : (X, τ ) −→ (Y, σ) ❧➭ ❘✲➳♥❤ ①➵ ✈➭ t✐Ị♥ ω ✲➤ã♥❣✱ t❤×

f (A) ❧➭ t❐♣ r❣ω ✲➤ã♥❣ tr♦♥❣ Y
❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

♠ë ❝❤Ý♥❤ q✉② ❝đ❛

Y

✈í✐ ♠ä✐ t❐♣ ❝♦♥ r❣

ω ✲➤ã♥❣ A ❝ñ❛ X ✳

A ❧➭ t❐♣ ❝♦♥ r❣ω ✲➤ã♥❣ ❜✃t ❦ú ❝ñ❛ X

s❛♦ ❝❤♦

f (A) ⊆ U ✳
✶✾

❉♦

f

✈➭

U

❧➭ ❘✲➳♥❤ ①➵ ♥➟♥

❧➭ t❐♣ ❝♦♥

f −1 (U )

❧➭


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 ✳ ❉♦ ó f



ợ

ị ý

ớ 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, ) r ó ế ỉ ế

ỗ t❐♣ ♠ë

B⊆V
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 ❝ñ❛

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 ❝đ❛ Y

❧➭ t❐♣ ❝♦♥ r❣

clω (V ) ❧➭ ω ✲➤ã♥❣ tr♦♥❣ Y ✱ ❞♦ ➤ã f −1 (clω (V )) ❧➭ t❐♣ ❝♦♥ r ó ủ X

ì

V

ợ

ó ❞♦ ➤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❐♣ ♥ư❛ ➤ã♥❣ ✭s❡♠✐ ❝❧♦s❡❞✮ ♥Õ✉ X − A ❧➭ t❐♣ ♥ư❛ ♠ë❀

✭❝✮

A

➤➢ỵ❝ ❣ä✐ ❧➭

ω ✲♥ư❛

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 ✲♥ư❛ ♠ë✳
❚❐♣ t✃t ❝➯ ❝➳❝ t❐♣

ω ✲♥ư❛ ♠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 ❤✐Ư✉ scl(A)

ét ỗ t ở t

ử ở ỗ t ử ở

t ử ở
❍ỵ♣ ❝đ❛ ❤ä t✉ú ý ❝➳❝ t❐♣
❧➭ t❐♣

ω ✲♥ư❛ ♠ë ❧➭ t❐♣ ω ✲♥ö❛ ♠ë✳ ❉♦ ➤ã sintω (A)

ω ✲♥ö❛ ♠ë❀
✷✶


✭❝✮ ●✐❛♦ ❝đ❛ ❤ä tï② ý ❝➳❝ t❐♣

ω ✲♥ư❛

➤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 ✲♥ư❛

➤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 ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ 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 ✲♥ö❛

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 ✲♥ư❛

♠➭

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✐ ❧➭ ♥ư❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ q✉② ✭s❡♠✐✲r❡❣✉❧❛r ❣❡♥❡r❛❧✲

✐③❡❞ ❝❧♦s❡❞✮ ♥Õ✉

scl(A) ⊂ U ✱ ✈í✐ ♠ä✐ t ở í q U

AU

ết

sró



ệ ề ỗ t ó t

ỗ t

só t ✲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 ♠➭


sử

A t só U

ì ỗ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❧➭ ♠ë ✈➭
❱❐②

❧➭ t❐♣ ♠ë ❝❤Ý♥❤ q✉② ❜✃t ❦ú ♠➭

A ⊂ U✳

A ❧➭ t❐♣ ω ❣s✲➤ã♥❣ t❛ s✉② r❛ sclω (A) ⊂ U ✳

A ❧➭ t❐♣ sró

ét ỗ t ở t

sở ỗ t sở t

srở



ệ ề ỗ t❐♣

ω ✲♥ö❛ ➤ã♥❣ s✉② ré♥❣ ❝❤Ý♥❤ 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❐♣ ❝♦♥ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ 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✳ ●✐➯ sư

❑❤✐ ➤ã

sclω (A) − A ❦❤➠♥❣ ❝❤ø❛ t ó í q rỗ ủ

t

X

ủ ❦❤➠♥❣ ❣✐❛♥ 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✳ ❑❤✐ ➤ã ♥❤ê ➜Þ♥❤ ❧ý ✷✳✶✳✾

F = φ ✈➭ ✈× 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✲❧✐➟♥ tô❝ ♥Õ✉

tô❝ s✉② rộ

ớ ỗ t ó

f



F

tr

(Y, ) t ó 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 tt sr

í q

tụ ế ớ ỗ t ➤ã♥❣

F

tr♦♥❣

(Y, σ) t❛ ❝ã f −1 (F ) ❧➭ t❐♣ ω ✲sr❣✲➤ã♥❣ tr♦♥❣

(X, τ )✳
✷✳✷✳✷

➜Þ♥❤ ❧ý✳ ●✐➯ sư

f : (X, τ ) −→ (Y, σ) ❧➭ ➳♥❤ ①➵✳

❑❤✐ ➤ã ❝➳❝ ❦❤➻♥❣

➤Þ♥❤ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣
✭❛✮

f

❧➭ ➳♥❤ ①➵

ω ✲sr❣✲❧✐➟♥ tơ❝❀

✭❜✮ ◆❣❤Þ❝❤ ủ ỗ t ở tr
ứ

(Y, ) 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➵ ❧✐➟♥ tô❝ t❤× ho f : (X, τ ) −→ (Z, δ) ❧➭ ➳♥❤

①➵

ω ✲sr❣✲❧✐➟♥ tơ❝✳
❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

❧✐➟♥ tơ❝ ♥➟♥

E

❧➭ t❐♣ ➤ã♥❣ ❜✃t ❦ú tr♦♥❣

h−1 (E) ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ (Y, σ)✳
✷✺

▲➵✐ ✈×

(Z, δ)✳

f

❧➭

❉♦

h ❧➭ ➳♥❤ ①➵

ω ✲sr❣✲❧✐➟♥ tơ❝✱ ♥➟♥