5
$\begingroup$

I know that in a compact Hausdorff space the quasi-components of the points coincide with their connected components, but I'd like to know if the Hausdorff hypothesis is needed.

I could not find a counterexample, i.e. a compact space in which a connected component is strictly contained in a quasi-component. I tried with the finite-complement topology, but it is connected. So it cannot be a counterexample because such a space must have infinite connected components.

So my question is: is there such a counterexample?

P.S. Sorry for my bad English, but I'm Italian and don't speak English very well.

| cite | improve this question | |
$\endgroup$
  • 1
    $\begingroup$ I suggest you provide more context. What do you know about the relation of a point's quasi-component and its connected component in general? How would it be defined without assuming Hausdorff separation? Etc. $\endgroup$ – hardmath Jan 30 '18 at 22:06
  • 1
    $\begingroup$ Thanks for the suggestion, i'll keep in mind for the future, and even though with a big delay, now i answer your question: i know that in general the components of points are included in the quasi-component, for the definitions, the quasi-components of x is the intersection of clopens containing x. $\endgroup$ – G. Ottaviano Feb 24 '18 at 15:40
  • $\begingroup$ Thanks for the reply! If I may suggest it, when you find an Answer like Eric's satisfactory, you can indicate this by clicking the Accepted check mark next to the Answer. $\endgroup$ – hardmath Feb 24 '18 at 15:48
4
$\begingroup$

Here's a simple example. Let $X=\mathbb{N}\cup\{a,b\}$, where a set $U\subseteq X$ is open iff $U\subseteq\mathbb{N}$ or $U$ is cofinite. The clopen subsets of $X$ are the finite subsets of $\mathbb{N}$ and the cofinite sets which contain both $a$ and $b$. In particular, $X$ is totally disconnected and compact, but $a$ and $b$ are in the same quasicomponent.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ +1 Thank you very much, this is a nice example. And sorry for the delay but in this period i was very busy. $\endgroup$ – G. Ottaviano Feb 24 '18 at 15:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.