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May 25

sufficient condition for two set to be sepreate-able (topology)

Hi, I look for a sufficient condition that for the next claim to hold;

Let A be a compact subset $\mathbb {R} ^{n}$
and let B1 and B2 two different connected component. So there are 2 Open subset $O_{1},O_{2}$

1.  $O_{1}\cup O_{2}\supseteq A$ 
2.  $cl(O_{1})\cap cl(O_{2})=\emptyset$ 
3.  $B_{1}\subseteq O_{1}$ 
4.  $B_{2}\subseteq O_{2}$

Is there supposed to be some relation between

A

and the pair

\{B_{1},B_{2}\}

, or are all three just given? --Lambiam 23:44, 25 May 2021 (UTC)[reply]

B1 and B2 are connected component of A.--Exx8 (talk) 05:14, 26 May 2021 (UTC)[reply]

No additional conditions are necessary. The statement is true as given.--2406:E003:855:9A01:74B0:C329:6D75:B8CD (talk) 06:49, 26 May 2021 (UTC)[reply]

Is it true that the connected components of a compact space are all open? If so, one can take

O_{1}=B_{1},

O_{2}=A\setminus B_{1}.

No. The connected components of the Cantor middle third set are the singletons.2406:E003:855:9A01:6D91:C1FE:E529:AA45 (talk) 00:00, 28 May 2021 (UTC)[reply]

@Exx8 Maybe this is equivalent to saying the components are all bounded away from each other- that is, given any two components

B_{1},B_{2}

there is some

\epsilon >0

such that

d(x,y)>\epsilon

whenever

x\in B_{1}

and

y\in B_{2}

Not equivalent (consider

\{(x,y)\in \mathbb {R} ^{2}\colon |xy|>1\}

) but the implication goes the right direction. --JBL (talk) 13:28, 28 May 2021 (UTC)[reply]