Limit Points of a subset are Limit Points of the original Set

Last updated Nov 1, 2022

Suppose $(X, \tau)$ is a Topological Space and let $T \subset S \subset X$. Then if $x \in X$ is a Limit Point of $T$, it is also a Limit Point of $S$.

Suppose $U \subset X$ Open so that $x \in U$. Then $U \cap S \supset U \cap T \neq \emptyset$ and $x$ is a Limit Point of $S$. $\blacksquare$