# Math4201 Topology I (Lecture 30)

## Compact and connected spaces

### Locally compact

#### Theorem of Homeomorphism over locally compact Hausdorff spaces

$X$ is a locally compact Hausdorff space if and only if there exists topological space $Y$ satisfying the following properties:

1. $X$ is a subspace of $Y$.
2. $Y-X$ has one point (usually denoted by $\infty$).
3. $Y$ is compact and Hausdorff.

