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The particular point topology on any set is connected, but on removing the particular point, the complement is discrete, and hence totally disconnected. Although this is not even $T^1$, Cantor's leaky tent also has this property, and is a subspace of $\mathbb{R}^2$.

Is there an infinite connected topological space such that the complement of every point is totally disconnected? Or in other words, is there an infinite connected space such that every proper subset of at least two points is disconnected?

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  • $\begingroup$ I erased my answer because I realized that I misread your question. Are you asking for $X$ connected such that for every $x\in X$, $X\setminus\{x\}$ is totally disconnected? $\endgroup$
    – Andrea Mori
    Jan 11, 2014 at 16:15
  • $\begingroup$ @AndreaMori Yes. Is the phrasing still ambiguous? $\endgroup$
    – ronno
    Jan 11, 2014 at 16:34

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A point like that is called a dispersion point and it is an immediate consequence of the answer to “Antisymmetry” among cut points that a connected space with at least three points cannot have more than one dispersion point.

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