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

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?

• 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? – Andrea Mori Jan 11 '14 at 16:15
• @AndreaMori Yes. Is the phrasing still ambiguous? – ronno Jan 11 '14 at 16:34