Is there an infinite connected topological space such that every space obtained by removing one point from it is totally disconnected?

general-topology examples-counterexamples connectedness

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?


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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