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.