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.

What I'm asking for is:

  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

  • 3
    $\begingroup$ 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. $\endgroup$ – Brian M. Scott Mar 3 '16 at 8:10
  • $\begingroup$ An example of a space with clopen connected components is the union of closed intervals in $\mathbb R$. $\endgroup$ – Math1000 Mar 3 '16 at 8:13
  • $\begingroup$ @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? $\endgroup$ – Dac0 Mar 3 '16 at 8:18
  • $\begingroup$ @Dac0: In the three specific examples that I gave I’m using the ordinary Euclidean topology inherited from $\Bbb R$. $\endgroup$ – Brian M. Scott Mar 3 '16 at 8:19
  • $\begingroup$ Now I got it! Thank You! Since they're dense with that topology the singleton is not open. Great example. $\endgroup$ – Dac0 Mar 3 '16 at 8:20

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