CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The one-point compactification of a topological space $X$ is a new compact space $X^+ = X \cup \{\infty\}$ obtained by adding a single new point “$\infty$” to the original space and declaring in $X^+$ the complements of the original closed compact subspaces to be open.
One may think of the new point added as the “point at infinity” of the original space. A continuous function on $X$ vanishes at infinity precisely if it extends to a continuous function on $X^+$ and literally takes the value zero at the point “$\infty$”.
This one-point compactification 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.
Let $X$ be any topological space. Its one-point extension $X^*$ is the topological space
whose underlying set is the disjoint union $X \cup \{\infty\}$
and whose open sets are
the open subsets of $X$ (thought of as subsets of $X^*$);
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.)
Dually in non-commutative topology the one-point compactification corresponds to the unitisation of C*-algebras.
$X^*$ 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.
The operation of one-point compactification is not a functor on the whole category of topological spaces. But it does extend to a functor on topological spaces with proper maps between them.
For $n \in \mathbb{N}$ the $n$-sphere (as a topological space) is the one-point compactification of the Cartesian space $\mathbb{R}^n$
Via this presentation of the $n$-sphere the canonical action of the orthogonal group $O(N)$ on $\mathbb{R}^n$ induces an action of $O(n)$ on $S^n$, which preserves the basepoint (the “point at infinity”).
This construction presents the J-homomorphism in stable homotopy theory and is encoded for instance in the definition of orthogonal spectra.
Slightly more generally, for $V$ any real vector space of dimension $n$ one has $S^n \simeq (V)^\ast$. In this context and in view of the previous case, one usually writes
for the $n$-sphere obtained as the one-point compactification of the vector space $V$.
For $V,W \in Vect_{\mathbb{R}}$ two real vector spaces, there is a natural homeomorphism
between the smash product of their one-point compactifications and the one-point compactification of the direct sum.
In particular, it follows directly from this that the suspension $\Sigma(-) \simeq S^1 \wedge (-)$ of the $n$-sphere is the $(n+1)$-sphere, up to homeomorphism:
For $X$ a compact topological space and $V \to X$ a vector bundle, then the (homotopy type of the) one-point compactification of the total space $V$ is the Thom space of $V$, equivalent to $D(V)/S(V)$.
For a simple example: the real projective plane $\mathbb{RP}^2$ is the one-point compactification of the ‘open’ Möbius strip, as line bundle over $S^1$. This is a special case of the more general observation that $\mathbb{RP}^{n+1}$ is the Thom space of the tautological line bundle? over $\mathbb{RP}^n$.