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On pg. 4 of this article on profinite groups is defined the following:

A topological space $X$ is said to be totally disconnected if each connected component of $X$ is a singleton.

Then the author remarks that a space $X$ is totally disconnected if and only if for any two distinct points in $X$, there is a closed and open subset of $X$ which contains one of the points but not the other.

I am unable to show this. One direction is trivial. But I am not able to conclude from the definition of total disconnectedness that one can "separate any two points by a disconnection."

rubik
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caffeinemachine
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    I believe this is false in general, the Knaster-Kuratowski fan (minus the top) is a counterexample, I believe. The fact is true for compact or locally compact Hausdorff spaces. – Henno Brandsma Aug 24 '16 at 14:35

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The statement is false in general. Two points of $X$ are in the same quasicomponent if there is no clopen set in $X$ containing exactly one of them. Thus, the quasicomponents are singletons if for any two points there is a clopen set containing exactly one of them. The component of a point is always a subset of the quasicomponent of the point, but the two need not be equal. They are always equal in compact Hausdorff spaces; this answer and this one contain proofs. In this answer I gave an example of a space in which they are not equal.

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Brian M. Scott
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  • Very interesting. I observed the same statement (regarding equivalence) on many web pages. Do you know if this slightly weaker disconnection property has a name? – AfterMath Jan 27 '21 at 15:04
  • @AfterMath: The property that distinct points can be separated by a clopen set, a sort of Hausdorff-like counterpart of zero-dimensionality? No, I don’t think that I’ve ever seen it given a name. – Brian M. Scott Jan 27 '21 at 20:10
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    @AfterMath: Such spaces are called totally separated. Sadly, there is inconsistent terminology in the literature. For instance, Engelking in "General Topology" calls a space "totally disconnected" if it is totally separated and if a space is totally disconnected (in the standard sense), he calls it "hereditarily disconnected." – Moishe Kohan Mar 01 '21 at 00:36
  • Incidentally, you gave another example in your earlier answer here. – Moishe Kohan Mar 01 '21 at 00:42