Khảo sát quang lực tác động lên vi cầu nhúng trong chất lưu nhiễu loạn

✶

▼Ö❈ ▲Ö❈
▼ö❝ ❧ö❝
▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶ ❍➺ ❤➔♠ ❧➦♣ ✈æ ❤↕♥ ✈➔ t➟♣ ❜➜t ❜✐➳♥
✶✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ❝ì sð

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷✳ ❍➺ ❤➔♠ ❧➦♣ ✈æ ❤↕♥ ✈➔ t➟♣ ❜➜t ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶

✷

✹
✹

✻

✶✳✸✳ ❑❤æ♥❣ ❣✐❛♥ ❞à❝❤ ❝❤✉②➸♥✱ →♥❤ ①↕ ❞à❝❤ ❝❤✉②➸♥ ✈➔ ♣❤➨♣ ❝❤✐➳✉
❝❤➼♥❤ t➢❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✷ ▼ët sè t➼♥❤ ❝❤➜t tæ♣æ ❝õ❛ t➟♣ ❜➜t ❜✐➳♥

✶✾

✷✳✶✳ ❚➼♥❤ ❧✐➯♥ t❤æ♥❣ ✈➔ ❧✐➯♥ t❤æ♥❣ ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✷✳✷✳ ❚➼♥❤ ❧✐➯♥ t❤æ♥❣ ✤÷í♥❣

❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹




é
ồ rt ữủ ữợ ỳ ỳ ừ t
ữớ t tr ỹ rt ố
tữủ ự ừ õ t rt ts
ữ r ởt ỹ t rt ỡ r
r ự õ ởt ồ ỳ tr Rn t ổ s r ởt
t rt ồ ồ õ trt t st
t t q s t rt ị
tữ s õ ữủ rs t tr sỷ ử
ởt rở r õ ộ ỹ rở ỵ tữ
ừ ts tr õ õ ữợ ởt t ổ

Rn ổ tr ổ tổổ rở ồ ỳ
t ổ t ự
trú ừ t rt ở sr ữớ t
ự t t tổổ ữ tổ tổ ữớ tổ
ữỡ ữ r ổ

t trt t st ự sỹ tỗ
t ụ ữ t t tổổ ừ t t q
t ữủ t ồ q t ự
t t ữủt ợ ự ồ t
ú tổ ồ t ừ ổ

t t tổổ ừ t t
ử ừ ỹ tr t t tr
ởt õ tố t q ự tt sỹ tỗ




t t t q ổ tr ổ tr ỗ
tớ tr ự tt ởt số t t tổổ ừ t t
t ử ồ ợ ử õ ở ữủ
tr tr ữỡ

ữỡ ổ t t
r ữỡ ú tổ tr tự ỡ s ũ

tr t ự tt sỹ tỗ t t t
ừ tr ổ tr

ữỡ ởt số t t tổổ ừ t t
r ữỡ ú tổ tr ự tt

t t tổổ t tổ tổ ữớ tổ
ữỡ ừ t t q ổ tr ổ
tr ử ồ

ữủ tỹ t trữớ ồ ữợ sỹ ữợ
t t ừ ổ ụ ỗ tọ ỏ t ỡ
s s ừ ổ t ỡ ừ ỏ
ồ ừ ữ t ồ qỵ t ổ
tr tờ t ừ rữớ ồ rữớ ồ ỏ
ú ù tr tớ ồ t r t
ỷ ớ ỡ rữớ
r ữỡ t t ủ
tr sốt tớ ồ t ố ũ ỡ
tr ợ ồ t ú ù ở tr sốt
q tr ồ t ự ũ õ ố s
ổ tr ọ ỳ t sõt rt ữủ
ỳ ỵ õ õ ừ qỵ ổ ữủ
t ỡ

ỗ t

ổ


✹

❈❍×❒◆● ✶

❍➏ ❍⑨▼ ▲➄P ❱➷ ❍❸◆ ❱⑨ ❚❾P ❇❻❚ ❇■➌◆
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ❝➛♥ ❞ò♥❣
tr♦♥❣ t♦➔♥ ❧✉➟♥ ✈➠♥✱ ❝❤ù♥❣ ♠✐♥❤ sü tç♥ t↕✐ t➟♣ ❜➜t ❜✐➳♥ q✉❛ ❤➺ ❤➔♠ ❧➦♣
✈æ ❤↕♥ ✈➔ ♠æ t↔ ❦❤æ♥❣ ❣✐❛♥ ❞à❝❤ ❝❤✉②➸♥✱ ♣❤➨♣ ❞à❝❤ ❝❤✉②➸♥ ✈➔ ♣❤➨♣ ❝❤✐➳✉
❝❤➼♥❤ t➢❝✳

✶✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ❝ì sð
✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✽❪✮ ❈❤♦ ❳ ❧➔ ♠ët t➟♣ ❤ñ♣ ❦❤→❝ ré♥❣✳ ▼ët →♥❤ ①↕

d : X × X → R t❤ä❛ ♠➣♥

✭✐✮ d(x, y)

0 ✈î✐ ♠å✐ x, y ∈ X ❀ d(x, y) = 0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y;

✭✐✐✮ d(x, y) = d(y, x) ✈î✐ ♠å✐ x, y ∈ X ❀
✭✐✐✐✮ d(x, z)

d(x, y) + d(y, z) ✈î✐ ♠å✐ x, y, z ∈ X ✳

❦❤♦↔♥❣ ❝→❝❤ ❤❛② ♠ët ♠➯tr✐❝ tr➯♥ X ✈➔ (X, d)
✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳

❑❤✐ ✤â✱ d ✤÷ñ❝ ❣å✐ ❧➔ ♠ët

✶✳✶✳✷ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✽❪✮ ❈❤♦ (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ (xn) ❧➔
♠ët ❞➣② tr♦♥❣ X ✳ ❉➣② (xn ) ✤÷ñ❝ ❣å✐ ❧➔

ε > 0, ∃ n0 ∈ N✱ s❛♦ ❝❤♦ ✈î✐ ∀n

❞➣② ❝ì ❜↔♥ ♥➳✉ ✈î✐ ♠å✐

n0 , ∀p ∈ N t❤➻ d(xn , xn+p ) < ε.

✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✶✵❪✮ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d) ✤÷ñ❝ ❣å✐ ❧➔ ✤➛② ✤õ
♥➳✉ ♠å✐ ❞➣② ❝ì ❜↔♥ tr♦♥❣ X ✤➲✉ ❤ë✐ tö✳

✶✳✶✳✹ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✶✵❪✮ ❈❤♦ X = ∅,

Ai

i∈I

❤å ❝→❝ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣

❝õ❛ X ✳ ❍å ♥➔② ✤÷ñ❝ ❣å✐ ❧➔ ❧✐➯♥ t❤æ♥❣ ♥➳✉ ✈î✐ ♠é✐ i, j ∈ I, tç♥ t↕✐ (ik )k=1,n ⊂

I s❛♦ ❝❤♦ i1 = i, in = j ♠➔ Aik ∩ Aik+1 = ∅ ✈î✐ ♠å✐ k ∈ {1, ..., n − 1}.


✺

✶✳✶✳✺ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✶✵❪✮✳ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d) ✤÷ñ❝ ❣å✐ ❧➔ ❧✐➯♥
t❤æ♥❣ ✤÷í♥❣ ♥➳✉ ✈î✐ ♠é✐ x, y ∈ X ❧✉æ♥ tç♥ t↕✐ ❤➔♠ ❧✐➯♥ tö❝ ϕ : [0, 1] → X

s❛♦ ❝❤♦ ϕ(0) = x ✈➔ ϕ(1) = y ✳

✶✳✶✳✻ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✶✵❪✮ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d) ✤÷ñ❝ ❣å✐ ❧➔ ❧✐➯♥ t❤æ♥❣
✤÷í♥❣ ✤à❛ ♣❤÷ì♥❣ ♥➳✉ ✈î✐ ♠é✐ x ∈ X ✱ ♠é✐ ❧➙♥ ❝➟♥ U

❝õ❛ x ✤➲✉ ❝â ♠ët

❧➙♥ ❝➟♥ ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣ ✤÷í♥❣ ♥➡♠ tr♦♥❣ U ✳

✶✳✶✳✼ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✶✵❪✮✳ ❍å
❣å✐ ❧➔

Ai

i∈I

❝→❝ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ X ✱ ✤÷ñ❝

❦❤æ♥❣ ❧✐➯♥ t❤æ♥❣ ♥➳✉ tç♥ t↕✐ J, J

⊂ I s❛♦ ❝❤♦ J ∪ J = I; J, J = ∅

s❛♦ ❝❤♦✿
Ai ✈➔ AJ , AJ

AJ ∩ AJ = AJ ∩ AJ = ∅, tr♦♥❣ ✤â✱ AJ =
i∈J

❧➔ ❜❛♦ ✤â♥❣ ❝õ❛ AJ , AJ t÷ì♥❣ ù♥❣✳
❍å Ai

i∈I

✤÷ñ❝ ❣å✐ ❧➔

❧✐➯♥ t❤æ♥❣ ♥➳✉ ♥â ❦❤æ♥❣ ❦❤æ♥❣ ❧✐➯♥ t❤æ♥❣✳

✶✳✶✳✽ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✶✵❪✮✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔
❧➔ ❤å ❝→❝ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ X ✳ ❍å Ai

t❤æ♥❣ ♠↕♥❤ ♥➳✉

i∈I

✤÷ñ❝ ❣å✐ ❧➔

Ai

i∈I

❦❤æ♥❣ ❧✐➯♥

∃J, J ⊂ I : J ∪ J = I; J, J = ∅, AJ ∩ AJ = ∅.

❍å Ai

i∈I

✤÷ñ❝ ❣å✐ ❧➔

❧✐➯♥ t❤æ♥❣ ②➳✉ ♥➳✉ ♥â ❦❤æ♥❣ ❦❤æ♥❣ ❧✐➯♥ t❤æ♥❣

♠↕♥❤✳

✶✳✶✳✾ ❈❤ó þ✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱

Ai

i∈I

❧➔ ❤å ❝→❝ t➟♣ ❝♦♥

❦❤→❝ ré♥❣ ❝õ❛ X ✳ ❚❛ ❝â
✶✮ ◆➳✉ Ai

i∈I

❦❤æ♥❣ ❧✐➯♥ t❤æ♥❣ ♠↕♥❤ t❤➻ ❝ô♥❣ ❦❤æ♥❣ ❧✐➯♥ t❤æ♥❣❀

✷✮ ◆➳✉ Ai

i∈I

❧✐➯♥ t❤æ♥❣ t❤➻ ❝ô♥❣ ❧✐➯♥ t❤æ♥❣ ②➳✉❀

✸✮ ◆➳✉ Ai

i∈I

❧✐➯♥ t❤æ♥❣ ♠↕♥❤ t❤➻ ❝ô♥❣ ❧✐➯♥ t❤æ♥❣❀

✹✮ ❚ç♥ t↕✐ ❤å ❧✐➯♥ t❤æ♥❣ ♥❤÷♥❣ ❦❤æ♥❣ ❧✐➯♥ t❤æ♥❣ ♠↕♥❤❀
✺✮ ❚ç♥ t↕✐ ❤å ❧✐➯♥ t❤æ♥❣ ②➳✉ ♥❤÷♥❣ ❦❤æ♥❣ ❧✐➯♥ t❤æ♥❣✳


✻

✶✳✶✳✶✵ ❇ê ✤➲✳ ✭❬✶✷❪✮✳ ❈❤♦ (X, dX ), (Y, dY ) ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔
→♥❤ ①↕ f : X → Y ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✱ Ai i∈I ❧➔ ❤å ❝→❝ t➟♣ ❝♦♥ ❦❤→❝
ré♥❣ ❝õ❛ X ✳ ❑❤✐ ✤â✱ t❛ ❝â
✶✮ ◆➳✉ ❤å Ai i∈I ❧✐➯♥ t❤æ♥❣ t❤➻ ❤å f (Ai) i∈I ❝ô♥❣ ❧✐➯♥ t❤æ♥❣❀

✷✮ ◆➳✉ Ai i∈I ❧✐➯♥ t❤æ♥❣ ②➳✉ t❤➻ ❤å f (Ai) i∈I ❝ô♥❣ ❧✐➯♥ t❤æ♥❣ ②➳✉✳
❈❤ù♥❣ ♠✐♥❤✳ ✶✮ ▲➜② J, J

⊂ I : J ∪ J = I, J ∩ J = ∅; J, J = ∅ ✈➔

AJ ∩ AJ = ∅ ❤♦➦❝ AJ ∩ AJ = ∅✳ ●✐↔ sû AJ ∩ AJ = ∅✳ ▲➜② a ∈ AJ ∩ AJ

t❤➻
f (a) ∈ f AJ ∩ AJ
f Ai ∩

=
i∈J

❱➟② f (Ai )

⊂ f AJ ∩ f AJ
f Ai

⊂ f AJ ∩ f AJ

= ∅.

i∈J

❧✐➯♥ t❤æ♥❣✳

i∈I

✷✮ ❱✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ t❤ü❝ ❤✐➺♥ t÷ì♥❣ tü✳

✶✳✷✳ ❍➺ ❤➔♠ ❧➦♣ ✈æ ❤↕♥ ✈➔ t➟♣ ❜➜t ❜✐➳♥
✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✶✵❪✮ ❈❤♦ (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔
x ∈ X, A ⊂ X ✳ ❑❤✐ ✤â✱ t❛ ✤à♥❤ ♥❣❤➽❛

❦❤♦↔♥❣ ❝→❝❤ tø ✤✐➸♠ x ✤➳♥ t➟♣ A❀
sup d(a, b) ✤÷ñ❝ ❣å✐ ❧➔ ✤÷í♥❣ ❦➼♥❤ ❝õ❛ A❀

✐✮ d(x, A) = inf d(x, a) ✤÷ñ❝ ❣å✐ ❧➔
✐✐✮ d(A) =

a∈A

a,b∈A

✐✐✐✮ d(A, B) = sup d(x, B) ✈î✐ A, B ⊂ X ❀
✐✈✮

K∗ (X)

x∈A

= {A : A ⊂ X, A ❝♦♠♣❛❝t✱ A = ∅};

P ∗ (X) = {A : A ⊂ X, A = ∅}.

✶✳✷✳✷ ▼➺♥❤ ✤➲✳ ✭❬✶✵❪✮ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝
✐✮ ⑩♥❤ ①↕ h✿

K∗ (X) × K∗ (X) → [0, +∞)

(A, B) → h(A, B) = max{d(A, B), d(B, A)}


✼

❧➔ ♠ët ♠➯tr✐❝ tr➯♥ K∗(X)✳
✐✐✮ ⑩♥❤ ①↕ h✿
P ∗ (X) × P ∗ (X) → [0, +∞)
(A, B) → h(A, B) = max{d(A, B), d(B, A)}

❧➔ ❣✐↔ ♠➯tr✐❝ tr➯♥ P ∗(X)✳
❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❚❛ ❦✐➸♠ tr❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ♠ët ♠➯tr✐❝ ✤è✐ ✈î✐ h✿
✰✮ ❚ø ❝→❝❤ ①→❝ ✤à♥❤ d(A, B) t❛ ❝â h(A, B)

0✱ ✈î✐ ♠å✐ A, B ∈ K ∗ (X)

✈➔ h(A, B) = 0 ❦❤✐ ✤â d(A, B) = d(B, A) ❤❛② A = B ❞♦✱ A, B ∈ K ∗ (X).
✰✮ ❍✐➸♥ ♥❤✐➯♥ h(A, B) = h(B, A)✳
✰✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤
h(A, B)

h(A, C) + h(C, B) ✈î✐ ♠å✐ A, B, C ∈ K∗ (X).

❚r÷î❝ ❤➳t t❛ ❝❤ù♥❣ ♠✐♥❤
d(A, B)

d(A, C) + d(C, B) ✈î✐ ♠å✐ A, B, C ∈ K∗ (X).

❚❛ ❝â
inf {d(a, c) + d(c, b)} ✈î✐ ♠å✐ c ∈ C.

d(a, B) = inf d(a, b)
b∈B

b∈B

=d(a, c) + d(c, B) ✈î✐ ♠å✐ c ∈ C.

❉♦ ✤â✱
d(a, B) = inf d(a, b)
b∈B

inf d(a, c) + inf d(c, b) = d(a, c) + d(c, B), ✈î✐ ♠å✐ c ∈ C.
b∈B

❱➟②✱ t❛ ❝â
d(A, B)

d(A, C) + d(C, B).

d(B, A)

d(B, C) + d(C, A).

❚÷ì♥❣ tü t❛ ❝â


✽

❉➝♥ ✤➳♥✱

h(A, B) = max{d(B, A), d(A, B))}
max{d(A, C) + d(C, B), d(B, C) + d(C, A)}
max{d(A, C), d(C, A)} + max{d(C, B), d(B, C)}
= h(B, C) + h(A, C).

❱➟②✱ h ❧➔ ♠ët ♠➯tr✐❝✳
✐✐✮ ✰✮ ❘ã r➔♥❣ A = B t❤➻ d(A, B) = d(B, A) = 0 ♥➯♥
max{d(A, B), d(B, A)} = 0

❤❛② h(A, B) = 0✳ ❱➟② h(A, A) = 0 ✈î✐ ♠å✐ A ∈ P ∗ (X)✳
✰✮ h(A, B)

0✱ ✈î✐ ♠å✐ A, B

✰✮ h(A, B) = h(B, A) ❧➔ ❤✐➸♥ ♥❤✐➯♥ ✈î✐ ∀A, B ∈ P ∗ (X)✳
✰✮ ❱✐➺❝ ❝❤ù♥❣ ♠✐♥❤ h(A, B)

h(A, C) + h(C, B) ✈î✐ ♠å✐ A, B, C ∈

P ∗ (X) ✤÷ñ❝ t✐➳♥ ❤➔♥❤ t÷ì♥❣ tü ♥❤÷ tr➯♥✳

❚✉② ♥❤✐➯♥✱ tç♥ t↕✐ A ∈ P ∗ (X) ♠➔ A = A ♥❤÷♥❣ h(A, A) = 0✳
❱➟②✱ h ❧➔ ♠ët ❣✐↔ ♠➯tr✐❝ tr➯♥ P ∗ (X)✳

✶✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✶✵❪✮ ⑩♥❤ ①↕ h ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ ð ▼➺♥❤ ✤➲ ✶✳✷✳✷

♠➯tr✐❝ ❍❛✉s❞♦r❢❢ ✈➔ ✤÷ñ❝ ❣å✐ ❧➔ ❣✐↔ ♠➯tr✐❝ ❍❛✉s❞♦r❢❢ ♥➳✉
♥â ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ ð ▼➺♥❤ ✤➲ ✶✳✷✳✷ ✭✐✐✮✳
✶✳✷✳✹ ▼➺♥❤ ✤➲✳ ✭❬✶✵❪✮✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ h ♥❤÷ ð
▼➺♥❤ ✤➲ ✶✳✷✳✷ ✭✐✐✮✳ ❑❤✐ ✤â✱ t❛ ❝â

✐✮ ◆➳✉ H, K ⊂ X, H, K = ∅ t❤➻ h(H, K) = h(H, K);
✐✐✮ ◆➳✉ Hi i∈I ✈➔ Ki i∈I ❧➔ ❝→❝ ❤å t➟♣ ❝♦♥ tr♦♥❣ X ✈➔ ❦❤→❝ ré♥❣
❝õ❛ X t❤➻

✭✐✮ ✤÷ñ❝ ❣å✐ ❧➔

h(H, K) = h

Hi ,
i∈I

Ki
i∈I

sup h(Hi , Ki );
i∈I

✐✐✐✮ ◆➳✉ H, K = ∅ ❝õ❛ X ✈➔ f : X → X ❧➔ →♥❤ ①↕ ▲✐♣s❝❤✐t③ t❤➻
h(f (K), f (H))

Lip(f ).h(H, K).






ổ tr ừ t
(K (X), h) ổ tr ừ (X, d) ổ
tr t t (K(X), h) ụ ổ tr t

(X, d)

(X, d) ổ tr f : X X.

õ

số st ừ f
x=y,x,yX
Lip(f ) < + t f ữủ ồ st
Lip(f ) < 1 t f ữủ ồ

Lip(f ) =

d(f (x),f (y))
d(x,y)

sup

ữủ ồ

ỵ (X, d) ổ
tr ừ f : X X õ ổ tỗ t
t x0 X f (x0) = x0 n
lim f n (x) = x0 ợ t x X.

(X, dX ) (Y, dY ) ổ
tr C(X, Y ) = {f : X Y |f tử } ồ

(fi )iI C(X, Y ) ữủ ồ

ợ ộ t A X

fi (A) tr Y.

t t
iI

ởt ổ tr ổ

tr ừ X ởt ồ (fi )iI , I t ổ
tr X s sup Lip(fi ) < 1 ỵ S = (X, (fi )iI )
iI

I t ỳ t S = (X, (fi )iI ) = (X, (fi )i=1,n ) ồ


ử sỷ (X, d) ổ tr ừ A t
õ rộ tr X A õ ỹ ữủ ổ ợ ộ
a A t
fa : X X
x fa (x) = a.


✶✵

❑❤✐ ✤â✱ fa ❧➔ →♥❤ ①↕ ❝♦❀ (fa )a∈A ❧➔ ❜à ❝❤➦♥ ✈➔ sup Lip(fa ) < 1✳ ❉♦ ✤â
a∈A

S = X, (fa )a∈A ❧➔ ✭■■❋❙✮✳

✷✮ ●✐↔ sû X = R2 ✈î✐ d ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❊✉❝❧✐❞✳ ❱î✐ ♠é✐ ✤÷í♥❣ t❤➥♥❣
d✱ t❛ ①➨t →♥❤ ①↕ πd : R2 → d ❧➔ ♣❤➨♣ ❝❤✐➳✉ tr➯♥ d✳ ❱î✐ ✷ ✤✐➸♠ A, B ❜➜t

❦➻ tr♦♥❣ R2 ✱ ❦➼ ❤✐➺✉ AB ❧➔ ✤÷í♥❣ t❤➥♥❣ q✉❛ ❤❛✐ ✤✐➸♠ A, B ✈➔ [A, B] ❦➼
❤✐➺✉ ❧➔ ✤♦↕♥ t❤➥♥❣ ❝â ❤❛✐ ✤➛✉ ♠ót ❧➔ A, B.
✣➸ þ r➡♥❣✱ ✈î✐ A, B, C ∈ R2 ♠➔ A = B ✱ ❧✉æ♥ tç♥ t↕✐ ❞✉② ♥❤➜t xC ∈ R
✤➸ πd (C) = A + xC (B − A) ✭t❛ ❤✐➸✉ B − A ❧➔ ♠ët ✤✐➸♠ ❝â tå❛ ✤ë tr♦♥❣
❦❤æ♥❣ ❣✐❛♥ R2 ❧➔ (xB − xA ; yB − yA )✮✳ ❉♦ ✤â✱ ✈î✐ A, B ∈ R2 , α ∈ R✱
①➨t →♥❤ ①↕
α
fA,B
: R2 → AB, ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐

 A ♥➳✉ xC < 0;
α
C → fA,B (C) =
A + xC .α.(B − A) ♥➳✉ xC ∈ [0; 1);

A + α(B − A) ♥➳✉ xC 1.
α
α ) =| α |✳
❘ã r➔♥❣✱ fA,B
❧➔ →♥❤ ①↕ ▲✐♣s❝❤✐t③ ✈î✐ Lip(fA,B
1

1

2
2
([A, B]) ∪ fB,A

([A, B]) = [A, B].
❑❤✐ α = 12 ✱ t❛ ❝â fA,B
n−1
1 1+(−1)
;
2n
2
fn ; f−2 ; f−1 : R2 →

❳➨t t➟♣ ❝→❝ ✤✐➸♠ An =
(0; 1) ✈➔ ❝→❝ ❤➔♠

✈î✐ n ∈ N ✈➔ A−1 = (0; 0), A−2 =

R2 ✤÷ñ❝ ❝❤♦ ❜ð✐


 (0, 0) = A−1 ♥➳✉ xC < 0
1
2
f−1 (C) =fA−1 ,A−2 (C) =
(0; 21 xc ) ♥➳✉ xC ∈ [0, 1)

(0, 12 ) ♥➳✉ xC 1,

 (0, 1) ♥➳✉ xC < 0
1
2
f−2 (C) =fA−2 ,A−1 (C) =
(0; 1 − 12 xC ) ♥➳✉ xC ∈ [0, 1)


(0, 21 ) ♥➳✉ xC 1,
1
2

fn (C) =

f2k = fAk ,Ak+1
1

f2k+1 = fA2 k+1 ,Ak


✶✶

❚❛ ❝â
1
2

f2k (C) = fAk ,Ak+1 (C) =

k−1





1 1+(−1)
♥➳✉ xC < 0
2

2k ;
k
1+(x
−1).(−1)
4−xC
C
;
♥➳✉ xC
2
2k+2
k+1
3 1+(−1)
♥➳✉ xC 1.
2
2k+2 ;




1

f2k+1 (C) = fA2 k+1 ,Ak (C) =







∈ [0, 1)

1+(−1)k
♥➳✉ xC > 0
2
2+xC 1+(1−xc ).(−1)k
♥➳✉ xC
2
2k+2 ;
k
1 1+(−1)
♥➳✉ xC 1
2
2k+1 ;
1

2k+1

;

∈ [0, 1)

❘ã r➔♥❣ fn , f−1 , f−2 ✤➲✉ ❧➔ ❝→❝ →♥❤ ①↕ ❝♦ ✈î✐ Lip(fn ) = Lip(f−1 ) =
Lip(f−2 ) =

1
2

✈➔ ❤å (fn )n

2

❤❛② (fn )n

✈➔ sup Lip(fi ) ❤❛② sup Lip(fi ) ✤➲✉
i −2

i 0

S1 = R2 , (fn )n

−1

−2 ✤➲✉ ❧➔ ❤å
❜➡♥❣ 12 ♥➯♥ S =

❝→❝ ❤➔♠ ❜à ❝❤➦♥

R2 , (fn )n

0

✈➔

✤➲✉ ❧➔ ❝→❝ ■■❋❙ tr➯♥ R2 .

✶✳✷✳✶✶ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✶✵❪✮ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d)✳ ❑➼ ❤✐➺✉ B(X)
❧➔ ❤å ❝→❝ t➟♣ ✤â♥❣✱ ❦❤→❝ ré♥❣✱ ❜à ❝❤➦♥ ❝õ❛ ❳ ✈➔ S = X, (fi

i∈I

) ❧➔ ■■❋❙

tr➯♥ X ✳ ❍➔♠ FS ✿
B(X) → B(X)
B → FS (B) =

fi (B)
i∈I

✤÷ñ❝ ❣å✐ ❧➔

t♦→♥ tû ❢r❛❝t❛❧✳

✶✳✷✳✶✷ ▼➺♥❤ ✤➲✳ ✭❬✶✷❪✮ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ S = (X, (fi)i∈I )

❧➔ ■■❋❙✳ ❑❤✐ ✤â✱ t♦→♥ tû ❢r❛❝t❛❧ FS ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ✣à♥❤ ♥❣❤➽❛
✶✳✷✳✶✶ t❤ä❛ ♠➣♥ Lip(FS ) sup Lip(fi)✳
i∈I


✶✷

❈❤ù♥❣ ♠✐♥❤✳ ▲➜② A, B ∈ B∗(X)✱ t❛ ❝â✿
h(F (A), F (B)) =h

fi (A),
i∈I

fi (B)
i∈I

=h

fi (A),
i∈I

fi (B)
i∈I

sup h(fi (A), fi (B))
i∈I

sup Lip(fi ).h(A, B) = c.h(A, B) ✈î✐ c < 1.
i∈I

❉♦ ✤â✱ S ❧➔ →♥❤ ①↕ ❝♦ ✈î✐ Lip(S)

c = sup Lip(fi ) < 1.
i∈I

●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈➔ S = (X, (fi )i∈I ) ❧➔ ■■❋❙
t❤➻ Lip(FS ) < 1 ♥➯♥ FS ❧➔ →♥❤ ①↕ ❝♦ ✈➔ (B(X), h) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr➼❝
✤➛② ✤õ✳ ❚❤❡♦ ♥❣✉②➯♥ ❧þ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤ t❛ ❝â ✤à♥❤ ❧þ s❛✉✳

✶✳✷✳✶✸ ✣à♥❤ ❧þ✳ ✭❬✶✷❪✮ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈➔ S =

❧➔ ■■❋❙ ✈î✐ c = sup Lip(fi) < 1✳ ❑❤✐ ✤â✱ tç♥ t↕✐ A ∈ B(X)
i∈I
s❛♦ ❝❤♦ FS (A) = A✳ ❍ì♥ ♥ú❛✱ ✈î✐ ❜➜t ❦➻ H0 ∈ B(X)✱ ❞➣② (Hn)n 0 ✤÷ñ❝
①→❝ ✤à♥❤ ❜ð✐ Hn+1 = FS (Hn) ❤ë✐ tö ✈➲ A t❤❡♦ ♠➯tr✐❝ ❍❛✉s❞♦r❢❢ ✈➔ tè❝

✤ë ❤ë✐ tö ✤÷ñ❝ ÷î❝ t➼♥❤ ❧➔
(X, (fi )i∈I )

cn
h(H0 , H1 ).
1−c

h(Hn , A)

✶✳✷✳✶✹ ◆❤➟♥ ①➨t✳ ❚r♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✾ ①➨t ❤➺ ❤➔♠ ❧➦♣ ✈æ ❤↕♥ ♥➳✉

c = sup Lip(fi ) = 1 t❤➻ ❦❤æ♥❣ ❝â t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ t➟♣ ❜➜t ❜✐➳♥ ✤÷ñ❝
i∈I

①→❝ ✤à♥❤ tr♦♥❣ ✣à♥❤ ❧þ ✶✳✷✳✶✸✳ ❚❤➟t ✈➟②✱ ①➨t ❤➺ ❤➔♠ ❧➦♣ ✈æ ❤↕♥ ✈î✐ n
fn : R → R
1
x → (1 − )x.
n

❑❤✐ ✤â✱ ❧➜② a, b ∈ R ♠➔ a
FS ([a, b]) =

0

b✱ t❛ ❝â

fn ([a, b]) =
n 1

1
1
[(1 − )a; (1 − )b] = [a, b].
n
n

n 1

1


✶✸

◆❤÷ ✈➟②✱ A = [a, b] ❧➔ t➟♣ ❜➜t ❜✐➳♥ ❝õ❛ ❤➺ ❤➔♠ ❧➦♣ ✈æ ❤↕♥✳ ❚✉② ♥❤✐➯♥ A
❧↕✐ ❦❤æ♥❣ ❞✉② ♥❤➜t ✈➔ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② t❛ ❝â Lip(fi ) = 1 −

1
n

♥➯♥

c = sup Lip(fi ) = sup(1 − n1 ) = 1.
n 1

i 1

✶✳✷✳✶✺ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✶✷❪✮ ❚➟♣ A tr♦♥❣ ✣à♥❤ ❧þ ✶✳✷✳✶✸ ✤÷ñ❝ ❣å✐ ❧➔ t➟♣
❜➜t ❜✐➳♥

❤❛② t➟♣ ❤ót ✭❛ttr❛❝t♦r✮✱ t➟♣ ❋r❛❝t❛❧ ❝õ❛ ❤➺ ❤➔♠ ❧➦♣ ✈æ ❤↕♥

S = (X, (fi )i∈I ).

✶✳✷✳✶✻ ❱➼ ❞ö✳ ✶✮ ❱î✐ S = (X, (fi)i∈I ) ♥❤÷ ð ❱➼ ❞ö ✶✳✷✳✶✵ ✭✶✮ t❤➻
fa (A) = A = A ✭❞♦ A ✤â♥❣✮✳

fa (A) = a ♥➯♥

a∈A

❱➟②✱ A ❧➔ t➟♣ ❜➜t ❜✐➳♥ ❝õ❛ ■■❋❙ S = (X, (fa )a∈A )✳
✷✮ ❱î✐ ❝→❝ ■■❋❙ ❧➔ S = R2 , (fn )n

❤❛② S1 = R2 , (fn )n

0

❱➼ ❞ö ✶✳✷✳✶✵ ✭✷✮ t❤➻ t➟♣ ❜➜t ❜✐➳♥ ✤➲✉ ❧➔ A = [A−1 , A−2 ]∪(

−2

♥❤÷ ð

[An , An+1 ])✳

n 0

❚❤➟t ✈➟②✱
1
f−1 ([A−1 , A−2 ]) = [A−1 , A0 ] ✈î✐ A0 = (0, ),

2
f−2 ([A−1 , A−2 ]) = [A0 , A−2 ].

❉♦ ✤â✱ f−1 ([A−1 , A−2 ]) ∪ f−2 ([A−1 , A−2 ]) = [A−1 , A−2 ].
❱➻ A−1 ✈➔ A−2 ✤➲✉ ❝â x = 0 ♥➯♥ f2k ([A−1 , A−2 ]) =
Ak ✈➔ f2k+1 ([A−1 , A−2 ]) =

k

1 1+(−1)
2
2k+1 ;

❉♦ ♠å✐ C ∈ [An , An+1 ] ✤➲✉ ❝â xC ∈

◆➳✉ n ❧➫ t❤➻ An =

1
2n ; 1

⇒ An+1 =

=

= Ak+1 .
⊂ [0; 1) ♥➯♥

1
1
2n+1 ; 2n

4 − xc 1 + (xc − 1)(−1)k
;
⊂
2
2k+1
◆➳✉ n ❝❤➤♥ t❤➻ An = 21n ; 0 ⇒ An+1 =
f2k (c) =

k−1
1 1+(−1)
;
k
2
2

1 1 + (−1)k−1
1 1
;
;
(
; ) .
2
2k
2k+1 2
1
2n+1 ; 1 .

1
2n+1 ; 0

.

1
1
⇒[A1 , A2 ] = ( ; 1); ( ; 0) ;
2
4
1
1
[A2 , A3 ] = ( ; 0); ( ; 1) ;
4
8
1
1
[A3 , A4 ] = ( ; 1); ( ; 0) .
8
16


✶✹

❱➻ ∀C ∈

An , An+1 ✤➲✉ ❝â xC ∈ [0; 1] ♥➯♥ tø ❝→❝❤ ①→❝ ✤à♥❤ f2n (c) =
n −2

A0 (x0 ; y0 ) t❤➻
1
c

x0 = 21n − 2xn+2
∈ 21n − 2n+2
; 21n ,
xc (−1)n
1+(−1)n−1
+
∈ [0, 1)
y0 =
2
2

❉♦ ✤â f2n ([An , An+1 ]) ⊂

[An , An+1 ].
n 0

✶✳✸✳ ❑❤æ♥❣ ❣✐❛♥ ❞à❝❤ ❝❤✉②➸♥✱ →♥❤ ①↕ ❞à❝❤ ❝❤✉②➸♥ ✈➔ ♣❤➨♣ ❝❤✐➳✉
❝❤➼♥❤ t➢❝
✶✳✸✳✶ ❈→❝ ❦➼ ❤✐➺✉✳ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ t❛ ❦➼ ❤✐➺✉
B A = {f : A → B};
∗

∧ = ∧(B) = B N = {f : N ∗ → B} = {ω = ω1 ω2 ... : ωi ∈ B};
∗

∧n = ∧n (B) = B Nn = {f : Nn∗ → B} = {ω = ω1 ...ωn : ω1 , ..., ωn ∈ B};
∧∗ = ∧∗ (B) =

∧n (B);
n 1

[ω]m = ω1 ...ωm , ✈î✐ ω ∈ ∧n (n

m) ❤❛② ω ∈ ∧.

α ∈ ∧n (B) ✈➔ β ∈ ∧m (B) t❛ ❦þ ❤✐➺✉ αβ = α1 ...αn β1 β2 ...βm ;

❤❛② α ∈ ∧n (B) ✈➔ β ∈ ∧(B) t❛ ❦þ ❤✐➺✉ αβ = α1 ...αn β1 β2 ...;
∞

ds (α, β) =
k=1

1 − δαβkk
✈î✐
3k

δαβkk = 1
δαβkk = 0

♥➳✉ βk = αk ,
♥➳✉ βk = αk .

❚❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ds ❧➔ ♠➯tr✐❝ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ∧✳ ❉♦ ✤â✱ (∧, ds ) ❧➔
❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳

✶✳✸✳✷ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✼❪✮ ❈❤♦ ∧(I) = (I N , ds) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔
∗

✤÷ñ❝ ❣å✐ ❧➔

❦❤æ♥❣ ❣✐❛♥ ❞à❝❤ ❝❤✉②➸♥ ❦➳t ❤ñ♣ ✈î✐ ■■❋❙ S

▲➜②
Fi : ∧(I) → ∧(I)
ω → iω

= (X, (fi )i∈I )✳


✶✺

✤÷ñ❝ ❣å✐ ❧➔

♣❤➨♣ ❞à❝❤ ❝❤✉②➸♥ ♣❤↔✐ ✈➔ ❧➔ ❝♦ ✈î✐ t➾ sè ❝♦ ❜➡♥❣ 13 ✈➻ t❛ ❝â✿

ds = (Fi (α), Fi (β)) = 31 ds (α, β)✳

❳➨t ❤➔♠ R :
∧(I) → ∧(I)
ω → ω2 ω3 ...

t❤➻ ds (R(α), R(β)) = 3ds (α, β) − (1 − δαβii )

3ds (α, β)✳

❞♦ ✤â ds ❦❤æ♥❣ ❧➔ →♥❤ ①↕ ❝♦ ✈➻ Lip(R) = 3 ♥➯♥ ❝❤➾ ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✳

✶✳✸✳✸ ◆❤➟♥ ①➨t✳ ✐✮ ❚❛ ❝â
ds R(α), R(β) = 3ds (α, β) − (1 − δαβ11 )

3ds (α, β)

✈➔ R ❝ô♥❣ ❧➔ ❤➔♠ ❧✐➯♥ tö❝✳
✐✐✮ R ◦ Fk (ω) = ω ✈➔ Fk ◦ R(ω) = kω2 ω3 ...
✐✐✐✮ ❑❤✐ B = I ❧➔ t➟♣ ❝❤➾ sè ✈æ ❤↕♥ ♥➔♦ ✤â t❤➻ ∧(I) =

Fk (∧(I)) ✈➔ ❞♦
k∈I

✤â ∧(I) ❝❤➼♥❤ ❧➔ t➟♣ ❜➜t ❜✐➳♥ ❝õ❛ ❤➺ ❤➔♠ ❧➦♣ ✈æ ❤↕♥ δ = ∧(I), (Fk )k∈I ✳

✶✳✸✳✹ ❈→❝ ❦➼ ❤✐➺✉✳ ✶✮ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ S = (X, (fi)i∈I ) ❧➔

■■❋❙ tr➯♥ X ✈î✐ ω = ω1 ω2 ...ωm ∈ ∧m (I) ⇒ fω = fω1 ...fωm ✈➔ Hω =
fω (H)✳

✷✮ ❈❤♦ ❤➔♠ f : X → X ❧➔ →♥❤ ①↕ ❝♦✳ ❑➼ ❤✐➺✉ ef ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣
❝õ❛ f. ◆➳✉ f = fω t❤➻ ❦➼ ❤✐➺✉ eω t❤❛② ❝❤♦ efω.
✣à♥❤ ❧þ s❛✉ ✤➙② ❝❤♦ t❛ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❦❤æ♥❣ ❣✐❛♥ ❞à❝❤ ❝❤✉②➸♥ ✈➔
t➟♣ ❜➜t ❜✐➳♥ ❝õ❛ ■■❋❙✳

✶✳✸✳✺ ✣à♥❤ ❧þ✳ ✭❬✺❪✮✳ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✱ S =
(X, (fi )i∈I ) ❧➔ ■■❋❙ ✈î✐ c = sup Lip(fi ) < 1 ✈➔ A ❧➔ t➟♣ ❜➜t ❜✐➳♥ q✉❛ ■■❋❙✳
i∈I

❑❤✐ ✤â✱ t❛ ❝â ❦❤➥♥❣ ✤à♥❤ s❛✉
✐✮ ❱î✐ ♠é✐ m ∈ N t❤➻ A[ω]
0✳ ❍ì♥ ♥ú❛✱ d(A[ω] ) = d(A[ω]

m+1

m

m

)

⊂ A[ω]m

✈î✐ ∀ω ∈ A ✈➔ m→∞
lim d(A[ω]

cm d(A).

m

)=


✶✻

✐✐✮ ◆➳✉ {aω } =

t❤➻ m→∞
lim d(e[ω

A[ω]m

m]

m∈N ∗

, aω ) = 0.

✐✐✐✮ ❱î✐ ♠é✐ a ∈ A ✈➔ ♠é✐ ω ∈ ∧ t❛ ❝â m→∞
lim f[ω]
✐✈✮ ❱î✐ ♠é✐ α ∈ ∧∗ t❛ ❝â A = {aω } ✈➔ Aα =
♥➳✉ A =

fi (A)

ω∈∧

t❤➻ A =

m

(a) = aω ✳
{αaω }✳

❍ì♥ ♥ú❛✱

ω∈∧

{aω }.
ω∈∧

i∈I

❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❱î✐ m = 1✱ t❛ ❝â A[ω]
→♥❤ ①↕ ❝♦✳ ❱î✐ m

1

= fw1 (A) ⊆ A ✈➻ fω1 ∈ {fi }i∈I ❧➔

2✱ t❛ ❝â

A[ω]m = f[ω]m (A) = f[ω]m−1 ◦ fω (A)

❚❛ ❝â d(A[ω]m ) = d(A[ω]m )

f[ω]m−1 (A) = A[ω]m−1 .

cm d(A) ✈î✐ ♠å✐ m ∈ N ✳
A[ω]m ❝❤➾ ✤ó♥❣ ♠ët ♣❤➛♥ tû✳

✐✐✮ ❉♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤➛② ✤õ ♥➯♥
m∈N ∗

●✐↔ sû ❣å✐ ♣❤➛♥ tû ✤â ❧➔ aω ✱ ♥❣❤➽❛ ❧➔ {aω } =

A[ω]m .
m>0

❱➻ f[ω]m ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ ✈î✐ ♠é✐ m > 0 ♥➯♥ t❛ ❝â
f[ω]m A[ω]m

f[ω]m (A) = A[ω]m

♥➯♥ t❛ ❝â e[ω]m ∈ A[ω]m ✳ ❉♦ ✤â✱ t❛ ❝â
d(e[ω]m , aω )

d(A[ω]m ) = d(A[ω]m )

cm d(A).

❱î✐ ♠é✐ m ∈ N∗ ✳ ❉♦ ✤â✱ lim d(e[ω]m , aω ) = 0.
m→∞

✐✐✐✮ ❱➻ d(f[ω]m (a), aω )

d(A[ω]m )

cm d(A).

❱î✐ ♠é✐ m > 0 ✈➔ lim cm = 0 ❞♦ 0
m→∞

c < 1 ♥➯♥ t❛ s✉② r❛ ✈î✐ ♠é✐

a ∈ A ✈➔ ♠é✐ ω ∈ A t❛ ❝â✱

lim f[ω]m (a) = aω .

m→∞

✐✈✮ ❘ã r➔♥❣ t❛ ❝â

{aω } ⊆ A✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❜❛♦ ❤➔♠ t❤ù❝ ♥❣÷ñ❝
ω∈∧

❧↕✐✱ tr÷î❝ ❤➳t t❛ ❝❤ù♥❣ ♠✐♥❤
Aω = A.
ω∈∧m


✶✼

❱î✐ ♠é✐ m ∈ N∗ ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ♥➔② ❜➡♥❣ q✉② ♥↕♣✱
❱î✐ m = 1✱ t❤❡♦ ✣à♥❤ ♥❣❤➽❛ t❛ ❝â A =

Aω .
ω∈∧m

●✐↔ sû ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ✤ó♥❣ ✈î✐ m✱ ♥❣❤➽❛ ❧➔
Aω = A.
ω∈∧m

❚❛ ❝❤ù♥❣ ♠✐♥❤

Aω = A.
ω∈∧m+1

❚❤➟t ✈➟②✱ ✈î✐ a ∈ A, ∃ ❞➣② (ai )i ❝→❝ ♣❤➛♥ tû ❝õ❛ Aωi ✈î✐ ωi ∈ ∧m s❛♦
❝❤♦ lim ai = a.
i→∞

❱➻ Aωi = fωi (A) = fωi

fωi (Aj ) ♥➯♥ ✈î✐ i ✤➣ ❝❤å♥ ❧✉æ♥

Aj ⊆
j∈I

j∈I

∃ji ∈ I ✈➔ bi ∈ (Aji )ωi s❛♦ ❝❤♦
1
d(ai , bi ) < .
i

❉♦ ✤â✱ lim bi = a ✈➔ bi ∈ ∧m+1 ✱ ❞♦ ✤â a ∈
i

Aω ✳
ω∈∧m+1

❱➟② A =

Aω ✳
ω∈∧m

❇➙② ❣✐í✱ t❛ ❝❤ù♥❣ ♠✐♥❤ A ⊆

{aω }.
ω∈∧

❚❤➟t ✈➟②✱ ✈î✐ a ∈ A, ✈➻ A =

Aω ♥➯♥ tç♥ t↕✐ ωm ∈ ∧m ✈➔ am ∈
ω∈∧m

Aωm s❛♦ ❝❤♦

d(am , a) <

1
.
m

✈î✐ α ∈ ∧✱ t❛ ❝â aωm α ∈ Aωm ✳ ❉♦ ✤â✱
d(am , aωm α)

d(A[ω]m ) = d(A[ω]m )

❉➝♥ ✤➳♥✱ lim aωm α = a ❤❛② a ∈
m→∞

{aω }.
ω∈∧

cm d(A).


✶✽

❱î✐ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝ô♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝
{aαω } ✈î✐ ♠é✐ α ∈ ∧∗ .

Aα =

ω∈∧

◆➳✉ A =

fi (A) t❤➻
i∈I

Aα = fα (A) = fα

fi (A) =
i∈I

Aαi ,
i∈I

✈î✐ ♠é✐ α ∈ ∧∗ ✳ ❉♦ ✤â✱ ✈î✐ a ∈ A✱ tç♥ t↕✐ ❞➣② (ω m )m ✈î✐ ❝→❝ t➼♥❤ ❝❤➜t
✭✶✮ ω m ∈ ∧m ;
✭✷✮ [ω m+1 ]m = ω m ;
✭✸✮ a ∈ Aωm ✈î✐ ♠å✐ m.
ω 1 = ω1 ✈➔ ω ω ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ω m = ω m−1 ωm t❤➻ a = aω ✈î✐
ω = ω1 .ω2 ∈ ∧. ❉♦ ✤â✱ a ∈

{aω }✳
ω∈∧

❱➟②✱ A ⊂

{aω }✳
ω∈∧

❱➻

{aω } = A✳

{a∧ } ⊆
ω∈∧

ω∈∧

❚❛ s✉② r❛ r➡♥❣

{aω } = A.
ω∈∧




ì

P ế P
r ữỡ ú tổ s tr ởt số t t ừ t
t q ổ tr ổ tr ữ t
tổ tổ ữớ tổ ữỡ ỗ tớ r sỹ
t t t tổổ ừ t t q s ợ t t
tổổ ừ t t q r ử ồ

tổ tổ

rữợ t t ờ s

ờ (X, d) ổ tr (Ai)iI ồ
tổ t tổ ừ X õ

Ai

tổ

iI

ự t tổ ừ t t q t
ỵ s

ỵ sỷ (X, d) ổ tr ừ

ợ c = max Lip(fi) < 1 A(S) t t
i=1,n
ừ õ s tữỡ ữỡ
ồ (Ai)i=1,n tổ ợ Ai = fi(A(S)) ợ ồ
A(S) tổ ữớ
A(S) tổ
ỵ (X, d) ổ tr ừ
S = (X, (fi )iI ) tr X sỷ (Ij )jJ I s
S = (X, (fi )i=1,n )


✷✵

✐✮ I =

Ij ;
j∈J

✐✐✮ Bj ❧➔ ❧✐➯♥ t❤æ♥❣ ✈î✐ Bj ❧➔ t➟♣ ❜➜t ❜✐➳♥ q✉❛ ❤➺ ❤➔♠ ❧➦♣
Sj = (X, (fi )i∈Ij ) ✈î✐ ♠å✐ j ∈ J;
✐✐✐✮ Bj ❧✐➯♥ t❤æ♥❣✳
j∈J

❑❤✐ ✤â✱ A ❧✐➯♥ t❤æ♥❣ ✈î✐ A ❧➔ t➟♣ ❜➜t ❜✐➳♥ q✉❛ S = (X, (fi)i∈I )✳
❈❤ù♥❣ ♠✐♥❤✳ ✣➦t C =

Bj ✈➔ (Cn )n ❧➔ ❞➣② ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✤÷ñ❝
j∈J

①→❝ ✤à♥❤ ❜ð✐ C0 := C; Cn := Fs (Cn−1 ) ✈î✐ n

fi (A)✳

1✱ ✈î✐ FS (A) =
i∈I

❚❛ t❤ü❝ ❤✐➺♥ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ q✉❛ ❤❛✐ ❜÷î❝✳

Cn ✈➔ ❜÷î❝ t❤ù

❇÷î❝ t❤ù ♥❤➜t✱ t❛ ❝❤ù♥❣ ♠✐♥❤ Cn ⊂ Cn+1 ✈➔ A =
n 1

❤❛✐ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣ r➡♥❣ Cn ❧➔ ❧✐➯♥ t❤æ♥❣ ✈î✐ ♠å✐ n✳ ❉♦

✤â✱ t❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ t➟♣ ❤ñ♣ ❧✐➯♥ t❤æ♥❣ t❛ ❝â A ❧✐➯♥ t❤æ♥❣✳
❚❛ ❝â✱ Bj = FSj (Bj ) ⊂ FS (Bj ) ✈î✐ ♠é✐ j ∈ J ✳ ❉♦ ✤â✱
Fs (Bj ) ⊂ Fs (

Bj ⊂

C = C0 =
j∈J

Bj ) = C1 .

j∈J

j∈J

❱➟②✱ C0 ⊂ C1 ✳ ❇➡♥❣ q✉② ♥↕♣✱ ❣✐↔ sû r➡♥❣ Cn ⊂ Cn+1 ❦❤✐ ✤â t❛ ❝â
Fs (Cj ) ⊂ Fs (Cn+1 ) ❤❛② Cn+1 ⊂ Cn+2 ✳

❱➟② t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ Cn ⊂ Cn+1 , ∀n
✣➦t D =

0.

Cn ✳ ❱➻ Cn = Fs (Cn−1 ) ✈î✐ ♠å✐ n

1 ♥➯♥ t❤❡♦ ✣à♥❤ ❧þ

n 1

✶✳✷✳✶✸ t❛ ❝â Cn → A✳

▲➜② n0 ∈ N∗ ✈➔ x ∈ Cn0 ✳ ❳➨t ❞➣② (xn )n
❱î✐ ♠é✐ n

n0

✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ xn = x1 ✳

n0 ✱ t❛ ❝â xn ∈ Cn ✈➻ Cn0 ⊂ Cn .

❉♦ ✤â✱ tø ▼➺♥❤ ✤➲ ✶✳✷✳✹ t❤➻ x = lim xn ∈ lim Cn = A✳ ❱➻ ❝❤å♥
n→∞

n→∞

x tò② þ tr♦♥❣ Cn0 ♥➯♥ t❛ ❝â Cn0 ⊂ A✳ ❉♦ ✤â✱ D =

Cn ⊂ A ✈➔
n 1

D ⊂ A = A ✈➻ A ✤â♥❣✳


✷✶

❇➙② ❣✐í✱ ❧➜② a ∈ A tø ▼➺♥❤ ✤➲ ✶✳✷✳✹✱ t❛ s✉② r❛ tç♥ t↕✐ (xn )n

1

s❛♦

Cn , ∀n ∈ N∗ ✈➔ xn → a ✭✤✐➲✉ ♥➔② ①↔② r❛ ✈➻

❝❤♦ xn ∈ Cn ⊂ D =
n 1

Cn → A✮✳ ❉♦ ✤â✱ A ⊂ D ✈➔ ✈➻ A ✤â♥❣ ♥➯♥ A = D.

❈✉è✐ ❝ò♥❣✱ t❛ ❝❤➾ r❛ r➡♥❣ Cn ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣ ✈î✐ ❜➜t ❦➻ n✳ ❚❛ s➩
❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ♥➔② ❜➡♥❣ q✉② ♥↕♣✳
❚❛ ❝â C0 = C ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣ t❤❡♦ ❣✐↔ t❤✐➳t✳ ●✐↔ sû r➡♥❣ Cn ❧✐➯♥
t❤æ♥❣ ✈î✐ ♠é✐ n > 0✳ ❚❛ ❝â✿
fi (Cn ) = C0

Cn+1 =

[

i∈I

fi (Cn )].

i∈I

❉♦ ✤â✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ Cn+1 ❧✐➯♥ t❤æ♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ C0
❧✐➯♥ t❤æ♥❣ ✈➻ ❜❛♦ ✤â♥❣ ❝õ❛ t➟♣ ❧✐➯♥ t❤æ♥❣ ❧➔ ❧✐➯♥ t❤æ♥❣✳

fi (Cn )
i∈I

❚❛ ❝❤ù♥❣ ♠✐♥❤✿ C0 ∩ fi (Cn ) = ∅ ✈î✐ ♠å✐ i ∈ I ✳

Ij ✳ ❑❤✐ ✤â✱ ❧✉æ♥ tç♥ t↕✐ l(i) ∈ J s❛♦ ❝❤♦ i ∈ Il(i) ✳ ❱➻

▲➜② i ∈ I =
j∈J

fi (A) ⊂ C0 ⊂ Cn ✳ ❉♦ ✤â✱ ✈î✐ ♠é✐ i ∈ I t❛ ❝â

Bl(i) = A(Sl(i) ) =
fi (Bl(i) ) ⊂ fi (Cn )✳

i∈Jl(i)

◆❤÷♥❣✱ fi (Bl(i) ) ⊂ FIl(i) (Bl(i) ) = Bl(i) ⊂ C0 ✳
❑❤✐ ✤â✱ φ = fi (Bl(i) ) ⊂ C0 ∩fi (Cn ) ✈î✐ ♠é✐ i ∈ I. ❱➟②✱ C0 ∩fi (Cn ) = ∅✳
❉♦ Cn ❧✐➯♥ t❤æ♥❣ ✈➔ fi ❧✐➯♥ tö❝ ♥➯♥ fi (Cn ) ❧✐➯♥ t❤æ♥❣✳
❉♦ ✤â✱
✤➲ ✷✳✶✳✶✳

fi (Cn ) ❧✐➯♥ t❤æ♥❣ ❤❛② Cn
i∈I

(

fi (Cn )) ❧✐➯♥ t❤æ♥❣ t❤❡♦ ❇ê

i∈I

❱➟②✱ Cn+1 = Cn

(

fi (Cn )) ❧✐➯♥ t❤æ♥❣✳

i∈I

◆❤÷ ✈➟②✱

Cn ✈➔ Cn ❧✐➯♥ t❤æ♥❣ ✈î✐ ♠é✐ n ∈ N✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦
n 0

Cn ❧➔ ❧✐➯♥ t❤æ♥❣ ✈➔ A = D ❧✐➯♥ t❤æ♥❣✳ ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣

D=
n 0

♠✐♥❤✳


✷✷

✷✳✶✳✹ ❍➺ q✉↔✳ ✭❬✸❪✮ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈➔
S = (X, (fi )i∈I )

✐✮ I =

❧➔ ■■❋❙✳ ▲➜② (Ij )j∈J ⊂ I s❛♦ ❝❤♦

IJ
j∈J

✐✐✮ (Bj )j∈J ❧✐➯♥ t❤æ♥❣✳
❑❤✐ ✤â✱ t➟♣ ❜➜t ❜✐➳♥ ❝õ❛ ■■❋❙ S = (X, (fi)i∈I ) ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣✳
❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❣✐↔ t❤✐➳t ✐✐✐✮ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✶✱ t❛ ❝â

Bj ❧✐➯♥ t❤æ♥❣
j∈J

♥➯♥ t❛ ❝â ❣✐↔ t❤✐➳t t❤ù ✸ ❝õ❛ ✣à♥❤ ❧þ ✷✳✶✳✸✳
❱➟②✱ tø ✣à♥❤ ❧þ ✷✳✶✳✸✱ t❛ ❝â t➟♣ ❜➜t ❜✐➳♥ ❧✐➯♥ t❤æ♥❣✳

✷✳✶✳✺ ❍➺ q✉↔✳ ✭❬✹❪✮ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈➔
S = (X, (fi )i∈I )

✐✮ I =

❧➔ ■■❋❙✱ ❣✐↔ sû (Ij )j∈J ⊂ I s❛♦ ❝❤♦

Ij .
j∈J

✐✐✮ Ij ❧➔ t➟♣ ❤ú✉ ❤↕♥ ✈î✐ ♠é✐ j ∈ J ✳
✐✐✐✮ ❍å ❝→❝ t➟♣ (fi(Bj ))i∈I ❧✐➯♥ t❤æ♥❣ ✈î✐ Bj ❧➔ t➟♣ ❜➜t ❜✐➳♥ q✉❛ ❤➺
❤➔♠ ❧➦♣ ❤ú✉ ❤↕♥ S = (X, (fi)i∈I ) ✈î✐ ♠é✐ j ∈ J.
✐✈✮ (Bj )j∈J ❧➔ ❤➺ ❧✐➯♥ t❤æ♥❣✳
❑❤✐ ✤â✱ t➟♣ ❜➜t ❜✐➳♥ A q✉❛ ❤➺ ❤➔♠ ❧➦♣ S = (X, (fi)i∈I ) ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣✳
j

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ ♠é✐ j ∈ J ✱ t❤❡♦ ✐✐✮ t❤➻ Ij ❤ú✉ ❤❛♥✳
❉♦ ✤â✱ S = (X, (fi )i∈I ) ❧➔ ❤➺ ❤➔♠ ❧➦♣ ❤ú✉ ❤↕♥ ♥➯♥ tø ❣✐↔ t❤✐➳t ✐✐✐✮ ✈➔
t❤❡♦ ✣à♥❤ ❧þ ✷✳✶✳✸ t❛ ❝â Bj ❧✐➯♥ t❤æ♥❣✳

❉♦ ✤â✱ ❣✐↔ t❤✐➳t ✐✐✮ ❝õ❛ ❍➺ q✉↔ ✷✳✶✳✺ ✤÷ñ❝ t❤ä❛ ♠➣♥✳
❚❤❡♦ ❍➺ q✉↔ ✷✳✶✳✺ t❛ s✉② r❛ A ❧✐➯♥ t❤æ♥❣✳

✷✳✶✳✻ ✣à♥❤ ❧þ✳ ✭❬✷❪✮ ●✐↔ sû (X, dX ) ✈➔ (Y, dY ) ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱
❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✈➔ (Ai)i∈I ❧➔ ❤å ❝→❝ t➟♣ ❦❤→❝ ré♥❣ tr♦♥❣ X.
✐✮ ◆➳✉ ❤å (Ai)i∈I ❧✐➯♥ t❤æ♥❣ t❤➻ ❤å f (Ai) i∈I ❝ô♥❣ ❧✐➯♥ t❤æ♥❣❀
✐✮ ◆➳✉ ❤å (Ai)i∈I ❧✐➯♥ t❤æ♥❣ ②➳✉ t❤➻ ❤å f (Ai) i∈I ❝ô♥❣ ❧✐➯♥ t❤æ♥❣
②➳✉✳

f :X→Y


✷✸

❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ▲➜② J, J

⊂ I s❛♦ ❝❤♦ J ∪ J = I, J ∩ J = ∅, J, J = ∅.

❱➻ ❤å (Ai )i∈I ❧✐➯♥ t❤æ♥❣ ♥➯♥ t❛ ❝â Ai ∩ AJ = ∅ ❤♦➦❝ AJ ∩ AJ = ∅
✈î✐ AJ =

Ai ✳ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❣✐↔ sû AJ ∩ AJ = ∅✳ ❑❤✐
i∈J

✤â✱ tç♥ t↕✐ a ∈ AJ ∩ AJ . ❚❛ ❝â✱
f (a) ∈ f (AJ ) ∩ f (AJ ) ⊂ f (AJ )∩f (AJ )
Ai ) ∩ f (

=f (
i∈J

=

i∈J

Ai ∩

i∈J

Ai )

Ai .
i∈J

❱➟②✱ tç♥ t↕✐ J, J ⊂ I, J ∪ J = I, J ∩ J =, J, J = ∅ ♠➔
f (Ai ) ∩
i∈J

❉♦ ✤â✱ ❤å f (Ai )

i∈I

f (Ai ) = ∅.
i∈J

❧✐➯♥ t❤æ♥❣✳ ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

✐✐✮ ❱✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ t❤ü❝ ❤✐➺♥ t÷ì♥❣ tü✳

✷✳✶✳✼ ❇ê ✤➲✳ ✭❬✷❪✮ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ

❧➔ ■■❋❙ ✈î✐ c = sup Lip(fi) < 1 ✈➔ A ❧➔ ♠ët t➟♣ ❜➜t ❜✐➳♥
i∈I
❝õ❛ ■■❋❙✳ ◆➳✉ ❤å (Ai)i∈I = (fi(A))i∈I ❧➔ ❧✐➯♥ t❤æ♥❣ ②➳✉ t❤➻ ❤å (Aω )ω∈∧
✈î✐ p ∈ N∗ ❧➔ ❧✐➯♥ t❤æ♥❣ ②➳✉✱ tr♦♥❣ ✤â
S = (X, (fi )i∈I )

p

∧p = ∧p (I) = {ω = ω1 ...ωp : ω1 , ..., ωp ∈ I}.

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣✳
❱î✐ p = 1 t❛ ❝â (Aω )ω∈∧1 = (Ai )i∈I ♥➯♥ t❤❡♦ ❣✐↔ t❤✉②➳t t❛ ❝â ✤✐➲✉
♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ✤ó♥❣ ❦❤✐ p = 1✳ ●✐↔ sû r➡♥❣ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ✤ó♥❣
✈î✐ p ❜➜t ❦➻✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ❤å (Aω )ω∈∧p+1 ❧✐➯♥ t❤æ♥❣ ②➳✉✳ ❑❤✐ ✤â✱ tç♥
t↕✐ J, J ⊂ ∧p+1 s❛♦ ❝❤♦
J ∪ J = ∧p+1 , J, J = ∅ ✈➔ AJ ∩ AJ = ∅.


✷✹

▲➜② ω ∈ ∧p ✱ ✤➦t
Jω = {ω ∈ J : [ω]p = ω} ✈➔ Jω = {ω ∈ J : [ω]p = ω}.

❑❤✐ ✤â✱ t❛ ❝â
Jω = ∅ ❤❛② Jω = ∅.

❚❤➟t ✈➟②✱ Jω ⊂ J ❀ Jω ⊂ J ✱ ❞♦ ✤â
AJω ∩ AJω ⊂ AJ ∩ AJ = ∅.

❱➻ ❤å (Ai )i∈I ❧➔ ❧✐➯♥ t❤æ♥❣ ②➳✉ t❤❡♦ ❣✐↔ t❤✐➳t ♥➯♥ tø ❇ê ✤➲ ✷✳✶✳✻ t❛
s✉② r❛ (Aω )ω∈Jω ∪Jω = (Aω )ω∈∧p+1 = (fω (Ai ))i∈I ❧➔ ❧✐➯♥ t❤æ♥❣ ②➳✉✳ ❉♦
✤â✱ tø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❧✐➯♥ t❤æ♥❣ ②➳✉ t❤➻ Jω = ∅ ❤♦➦❝ Jω = ∅.
▲↕✐ ✤➦t
L = {[ω]p : ω ∈ J};
L = {[ω]p : ω ∈ J }.

❚❛ ❝â L ∩ L = ∅; L = ∅; L = ∅; L ∪ L = ∧p ✈➔ AJ = AL
✈➔ AL ∩ AL = ∅.
✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ (Aω )ω∈∧p+1 ❦❤æ♥❣ ❧✐➯♥ t❤æ♥❣ ②➳✉✱ tr→✐ ✈î✐ ❣✐↔ t❤✐➳t✳
❉♦ ✤â✱ t❛ ❝â (Aω )ω∈∧p+1 ❧✐➯♥ t❤æ♥❣ ②➳✉✳

✷✳✶✳✽ ❇ê ✤➲✳ ✭❬✷❪✮ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ

❧➔ ■■❋❙ ✈î✐ c = sup Lip(fi) < 1 ✈➔ A ❧➔ t➟♣ ❜➜t ❜✐➳♥ ❝õ❛
i∈I
■■❋❙✳ ◆➳✉ ❤å (Ai)i∈I = fi(Ai∈I ❧➔ ❧✐➯♥ t❤æ♥❣ ②➳✉✱ ✈➔ A = fi(A) t❤➻
S = (X, (fi )i∈I )

❤å (Ai)i∈I ❧✐➯♥ t❤æ♥❣✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ♥❣÷ñ❝ ❧↕✐ ❤å (Ai)i∈I ❦❤æ♥❣ ❧✐➯♥ t❤æ♥❣✳
❑❤✐ ✤â✱ tç♥ t↕✐ J ✈➔ J ⊂ I ♠➔ J ∪ J = I, J, J = ∅
✈➔ AJ ∩ AJ = AJ ∩ AJ = ∅.(∗)
❱➻ ❤å (Ai )i∈I ❧✐➯♥ t❤æ♥❣ ②➳✉ ♥➯♥ t❛ ❝â AJ ∩ AJ = ∅.

i∈I

õ tỗ t x AJ AJ
ữ AJ AJ =

Ai = A(S) =
iI

Aj = AJ AJ

fj (A(S)) =
jI

jI

x AJ AJ tự x AJ x AJ
t AJ AJ = AJ AJ =
t ợ ự tọ ồ (Ai ) tổ

ỵ sỷ (X, d) ổ tr ừ
S = (X, (fi )iI )

ợ c = sup Lip(fi) < 1 A t t ừ
iI

õ
A t tổ t (Ai)iI ồ tổ ợ
Ai = fi (A), i I;

A tổ A =

fi (A)

t (Ai)iI ồ tổ

iI

ồ (Ai)iI tổ t A ổ t ữủ t
t tổ
ự A = Aw ợ w p p N t
rộ
sỷ ữủ ồ (Ai )iI ổ tổ õ tỗ t
J, J I s J J = I, J = , J = AJ AJ =

Ai = , AJ AJ = A =

Ai = AJ AJ
iI

t A ổ tổ r ợ tt t õ ự

ứ ờ t õ ự
sỷ ữủ r A ổ t ữủ tỗ
t t B, C s A = B C h(B, C) > 0 A õ B, C
õ
= h(B, C) ồ m số tỹ s C m d(A) <
1
m = [logc d(A)
] 1