(Theorem) Sequences in Hausdorff spaces converge to at most one point

Theorem (Munkres 17.8 & 17.10)

If $X$ is Hausdorff, then:

• (a) Every finite subset of $X$ is closed;
• (b) Every sequence in $X$ converges to at most one point of $X$.

Sketch.

1. Show (a) by showing every singleton set in a Hausdorff space is closed.
   • Method from Munkres: Since $X$ is Hausdorff, what does (Theorem) Describing the closure of a set using a topological basis tell us about the closure of a singleton set $\{x_0\}$?
   • Method from class: Use the Hausdorff property and Topology HW1, P1 to show $X\backslash \{x_0\}$ is open.
2. Show (b) by choosing disjoint neighborhoods for two distinct points, then showing that it is impossible fo a sequence to converge to both.