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

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

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

Thanks!-- Exx8 ( talk) 19:01, 25 May 2021 (UTC) reply

Is there supposed to be some relation between ${\displaystyle A}$ and the pair ${\displaystyle \{B_{1},B_{2}\}}$, 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

Can you prove it?-- Exx8 ( talk) 12:52, 26 May 2021 (UTC) reply

Is it true that the connected components of a compact space are all open? If so, one can take ${\displaystyle O_{1}=B_{1},}$ ${\displaystyle O_{2}=A\setminus B_{1}.}$  -- Lambiam 08:43, 27 May 2021 (UTC) reply

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 ${\displaystyle B_{1},B_{2}}$ there is some ${\displaystyle \epsilon >0}$ such that ${\displaystyle d(x,y)>\epsilon }$ whenever ${\displaystyle x\in B_{1}}$ and ${\displaystyle y\in B_{2}}$. Staecker ( talk) 11:32, 28 May 2021 (UTC) reply

Not equivalent (consider ${\displaystyle \{(x,y)\in \mathbb {R} ^{2}\colon |xy|>1\}}$) but the implication goes the right direction. -- JBL ( talk) 13:28, 28 May 2021 (UTC) reply