One-point compactification

Definition

Let $(X, \tau)$ be a topological space and let $\infty$ be a point not in $X$. Let $X^* = X \cup \{\infty\}$. The one-point compactification, or Alexandroff extension, of $X$ is the set $X^*$ with the following topology. The open sets are the sets $U \in \tau$ and the sets $X^* \setminus C$, where $C \subseteq X$ is closed and compact in $X$.

Remarks

The space $X^*$ is compact, and $X$ is an open subspace of $X^*$. The space $X^*$ is Hausdorff if and only if $X$ is locally compact and Hausdorff. The one-point compactification of $\mathbb{R}^n$ is homeomorphic to the sphere $S^n$, by stereographic projection. For $n = 1$, it is the circle.

Explanations

L1 human

You make a non-compact space compact by adding one point, and you declare that the new point is near everything outside large compact sets. The real line becomes a circle: both ends meet at the new point.