Exercise 10, § 2

Let $\overline{L}$ be the completed Alexandroff line and let S be the topological space obtained from $\overline{L}$ by identification of the points $-\infty$ and $+\infty$.

a) Prove that the space S is compact and connected, of cardinal at least 2.

b) Prove that for every pair $(x, y)$ of distinct points of S, the topological space S $-\{x, y\}$ is not connected.

c) Prove that the space S is not homeomorphic to a circle.