Connected subsets one of which meets the closure of the other
Exercise 224 (Chapter 32, “Connectedness”):
Let and be connected subsets of topological space such that intersects . Prove that is connected. Find an example to show that it is not enough to assume that intersects .
Let us assume that and are open subsets of such that is empty and that . To show that is connected, we need to show that either or does not intersect .
Since is connected, and is empty and , necessarily is entirely in or in . Similarly, is entirely in or entirely in .
Let us suppose that they are not both in or both in ; for instance, that and .
If , then with open in . Since does not intersect , cannot be in . Hence cannot intersect . This contradicts our assumption.
Hence it is impossible that and . Similarly, it is impossible that and . Thus, and are either both in or both in , which implies that either or does not intersect .
If we only assume that and intersect, it does not always follow that is connected. For instance, let be the topological plane and and two open disks that “almost touch”, that is such that the corresponding closed disks ‑ which are their closures ‑ intersect in one point. The union of and is then not connected.