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A topological space $X$ is totally disconnected if all of its components are singeltones, and $X$ is said to be totally separated if all of its quasi-components are singeltones. The set of rational numbers as well as the Cantor set are both totally disconnected and totally separated. I need an example of a space which is totally disconnected but not totally separated i.e. a space whose components are singeltones, but at least one of its quasi-components consist of more than one element.

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The standard example of such a space is Cantor's leaky tent. It has a single quasi-component, which is the whole space.

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