Exercise 8, § 2

Let X be a connected compact topological space of cardinal $\geqslant 2$. Suppose that for every pair $(x, y)$ of distinct points of X, the space X $\{x, y\}$ is not connected.

a) Prove that for every $x\in X$, the space X $-\{x\}$ is connected.

b) Let $a, b$ be distinct points of X and let $U,V$ be non-empty open and closed subsets of X such that X$-\{a, b\}= U\cup V$. Prove that one has $\overline{U}= U\cup \{a, b\},\overline{V}= V\cup \{a, b\}$ and that these sets are connected.

c) Suppose that X possesses a countable everywhere dense subset. Prove then that X is homeomorphic to the circle $\mathbf{S}_1$. (Prove that there exists a homeomorphism of $\mathbf{I}$ onto $\overline{U}$ which maps 0 onto $a$ and 1 onto $b.$)