One-Point Compactification

Definition

A Topological Space $(X,\mathcal{T})$ is Locally Compact Hausdorff Space if and only if there exists a Compact Hausdorff Space $Y\text{s.t.}Y=X\cup \{\infty \}$ where $x$ is a point. The $Y$ is called the one-point compactification of $X$.

Examples

One-point compactification of $\mathbb{R}$

One-point compactification of ${\mathbb{R}}^2$

Consider a Topological Space $(X,\mathcal{T})$ and its one-point compactification $(X_{\infty },{\mathcal{T}}_{\infty })$. $(X,T)$ is a Subspace Topology of $(X_{\infty },{\mathcal{T}}_{\infty })$.