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What is the relationship between discrete spaces and totally disconnected spaces, in general? Among Hausdorff spaces?

A space X is totally disconnected iff the only connected subsets of X are singletons, and A space X is discrete iff each point in X is an open set. That's all I know but I dont know how to relate this two. Can someone tell me any other main differences between the two other than these? I am totally unclear about the second question...Need help on that.

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  • $\begingroup$ Can you show that every discrete space is totally disconnected (and Hausdorff)? That tells you something about the relationship between the two properties. Finding a Hausdorff space that is totally disconnected but not discrete would tell you that the two really are different properties for Hausdorff spaces and hence also in general. What about the Cantor set? $\endgroup$ – Brian M. Scott Mar 24 '20 at 2:40
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    $\begingroup$ What about $\mathbb{Q}$? $\endgroup$ – Nuntractatuses Amável Mar 24 '20 at 3:21
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The only real relationship is that a discrete space is a special case of a totally disconnected space. But not every totally disconnected space is discrete, e.g. $\mathbb{Q}$ with the Euclidean topology is the standard example.

While discrete spaces are rather simple and well understood, totally disconnected can be quite complicated.

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