nLab
one-point compactification

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 $X\cup \{\mathrm{\infty }\}$

• 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.

Related concepts

• Thom space

• compactification

References

• John Kelly, General Topology (1975)