All points in a Set are Limit Points

Last updated Nov 1, 2022

Let $X, \tau$ be a Topological Space and let $S \subset X$. Then $$\bar{S} \supset S$$

Suppose $x \in S$. Suppose $U \subset X$ be Open s.t. $x \in U$. Then $U \cap S \supset {x}$. Since $U$ was arbitrary, $x$ is a Limit Point of $S$. $\blacksquare$