# nLab one-point compactification

### Context

#### Topology

topology

algebraic topology

# Contents

## Idea

The one-point compactification of a topological space is a new compact space obtained by adding a single new point to the original space.

This is also known as the Alexandroff compactification after a 1924 paper by Павел Сергеевич Александров (then transliterated ‘P.S. Aleksandroff’).

The one-point compactification is usually applied to a non-compact locally compact Hausdorff space. In the more general situation, it may not really be a compactification and hence is called the one-point extension or Alexandroff extension.

## Definition

Let $X$ be any topological space. Its one-point extension ${X}^{*}$ is the topological space

• whose underlying set is the disjoint union $X\cup \left\{\infty \right\}$

• and whose open sets are

1. the open subsets of $X$ (thought of as subsets of ${X}^{*}$);

2. the complements (in ${X}^{*}$) of the closed compact subsets of $X$.

(If $X$ is Hausdorff, then its compact subsets must always be closed, so (2) is often given in a simpler form.)

## Properties

${X}^{*}$ is compact.

The evident inclusion $X\to {X}^{*}$ is an open embedding.

The one-point compactification is universal among all compact spaces into which $X$ has an open embedding, so it is essentially unique.

$X$ is dense in ${X}^{*}$ iff $X$ is not already compact. Note that ${X}^{*}$ is technically a compactification of $X$ only in this case.

${X}^{*}$ is Hausdorff (hence a compactum) if and only if $X$ is already both Hausdorff and locally compact.

## References

• John Kelly, General Topology (1975)

Revised on April 11, 2013 02:12:17 by Todd Trimble (67.81.93.26)