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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.

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    $\begingroup$ 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, 2018 at 22:06
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    $\begingroup$ 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, 2018 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, 2018 at 15:48

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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.

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  • $\begingroup$ +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, 2018 at 15:02

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