Wikipedia:Reference desk/Archives/Mathematics/2021 May 25

= 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. $$B_{2}\subseteq O_{2}$$

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


 * 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)
 * B1 and B2 are connected component of A.--Exx8 (talk) 05:14, 26 May 2021 (UTC)
 * 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)
 * Can you prove it?--Exx8 (talk) 12:52, 26 May 2021 (UTC)
 * Is it true that the connected components of a compact space are all open? If so, one can take $$O_1=B_1,$$ $$O_2=A\setminus B_1.$$ --Lambiam 08:43, 27 May 2021 (UTC)
 * 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)
 * @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 $$y\in B_2$$. Staecker (talk) 11:32, 28 May 2021 (UTC)
 * Not equivalent (consider $$ \{(x, y) \in \R^2 \colon |xy| > 1 \}$$) but the implication goes the right direction. --JBL (talk) 13:28, 28 May 2021 (UTC)