When you cut something with a knife, does the blade come into contact with each of the little pieces? I think so.

Now consider a topological version of this question. Let be a connected topological space (with more than 1 point) which we “cut” by removing a point . Let be a connected component of ; this is one of the little pieces. *Is it true that ?*

**True** if we replace “connected” by “path-connected”. Indeed, let be a continuous function such that and . If necessary, we truncate so that for . Now the set is a path-connected subset of , hence . Since as , we have .

**False** as stated, even if is a subset of . Here is a counterexample given by Niels Diepeveen on Math.StackExchange:

The set consists of the point and of closed line segments from to . The union of line segments is connected, and since it is dense in , is also connected. However, once is removed, each line segment becomes its own connected component, and so does the point . Clearly, the closure of does not contain .

In this example is not compact. It turns out that the statement is true when is a compact connected Hausdorff space (i.e., a *continuum*), but the proof (also given by Niels Diepeveen) is not easy.

Have you heard of the Knaster-Kuratowski fan? It is a connected subset of the plane that becomes totally disconnected upon removal of a certain point. So the knife doesn’t touch any of the pieces that remain.

That’s a more impressive example, of course — but clearly beyond my ability to draw…