Limit Point

Last updated Nov 6, 2022

Suppose $(X, \tau)$ is a Topological Space. Let $S \subset X$. Then $x \in X$ is a Limit Point of $S$ if $\forall U \in \tau$ such that $x \in U$, we have that $U$ and $S$ are not Mutually Disjoint. That is, $U \cap S \neq \emptyset$.

We denote $$\bar{S} := {x \in X : x \text{ is a limit point of } S}$$

1. This notation should not be confused with the notation for the Closed Ball, $\overline{B_{r}(x)}$, which may contain points besides Limit Points. See Closure of Open Ball is a subset of Closed Ball

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