# How are related Connected Component with Open-Closed Subset

I used to think that connected components were closed and open at the same time, but I discovered quite recently (italian page of Wikipedia) that this might not be the case.

1. an example to illustrate why it is not the case (i.e. an example for which the connected component are not open);
2. if there's some kind of statement or condition which specifies when they are open and closed.

Thank You

• Let $X$ be any totally disconnected space; three familiar ones are $\Bbb Q$, $\Bbb R\setminus\Bbb Q$, and the Cantor set. Because these are totally disconnected, each component is a singleton; however, none of these spaces has an isolated point, so no singleton is open. – Brian M. Scott Mar 3 '16 at 8:10
• An example of a space with clopen connected components is the union of closed intervals in $\mathbb R$. – Math1000 Mar 3 '16 at 8:13
• @BrianM.Scott I'm not sure I got your example... please could you expand it a little bit? Which topology are you putting on $X$, the Zarinski topology? – Dac0 Mar 3 '16 at 8:18
• @Dac0: In the three specific examples that I gave I’m using the ordinary Euclidean topology inherited from $\Bbb R$. – Brian M. Scott Mar 3 '16 at 8:19
• Now I got it! Thank You! Since they're dense with that topology the singleton is not open. Great example. – Dac0 Mar 3 '16 at 8:20