Abhijeet Mulgund's Personal Webpage

Search

Search IconIcon to open search

Closure of a Set in a Metric Space is all its Sequential Limits

Last updated Nov 1, 2022

# Statement

Suppose $(X, d)$ is a Metric Space. Suppose $S \subset X$ Then $$\text{cl} S = {x : \exists (x_n) \subset S \text{ s.t. } x_{n} \to x}$$

# Proof

Metric Spaces are First Countable and Hausdorff and Closure of a Set in a First Countable Space is all its Sequential Limits. $\blacksquare$