Answer to Question #231846 in Differential Geometry | Topology for chanakya

Question

The real line with the usual topology is locally compact

Answer 1

$\text{$\mathbb{R}$ is locally compact since given $\epsilon > 0$ and x $\in \mathbb{R},$ the neighbourhood $(x-\epsilon, x+\epsilon )$}\\ \text{is contained in $[x-\epsilon, x+\epsilon ]$ and it is known that $[x-\epsilon, x+\epsilon ] \subset \mathbb{R}$ is compact. }\\ \text{Hence showing that $\mathbb{R}$ is locally compact.}$