added a remark (here) relating the example of Euclidean space compactifying to an $n$-sphere to discussion of monopoles and instantons in gauge theory. Also added a graphics illustrating this for charges in Cohomotopy theory.

]]>Thanks for the alert. I guess this must have been a typo for “locally compact Hausdorff”. I have edited accordingly. But check.

]]>The way the “Universal Property” section is written is confusing to me. It sounds like the universal property is only claimed when $X$ is a compact Hausdorff space – in which case the one-point compactification is not very useful!

]]>Internet Archive scan, not arXiv. I don’t think the arXiv would like giant scan of a published book uploaded!

]]>added pointer to what I guess is the original reference:

- Pavel Aleksandrov,
*Über die Metrisation der im Kleinen kompakten topologischen Räume*, Mathematische Annalen (1924) Volume: 92, page 294-301 (dml:159072)

and equipped the pointer to Kelly’s textbook with hyperlinks to the arXiv scan copy:

- John Kelly, p. 150 of:
*General Topology*, van Nostrand 1955 (archive:GeneralTopology)

I have added the remark here that every compact Hausdorff space is the one-point compactification of its complement by one point.

]]>I have made exlicit the proof that the one-point compactification of $\mathbb{R}^n$ is $S^n$ (here)

]]>Moreover, I have spelled out the proofs that:

if $X$ is locally compact then $X^\ast$ is Hausdorff precisely if $X$ is (here)

the inclusion $X \to X^\ast$ is an open embedding (here)

every locally compact Hausdorff space arises as an open subspace of a compact Hausdorff space (here)

The last statement had been at *locally compact space* without proof (elementary as it may be), and so I added pointer there.

just to be more boring still, I made explicit why $X^\ast$ is indeed compact: here

]]>just to bore everyone, I have written out in detail why the topology on the one-point extension is well defined in the first place: here

]]>I sometimes get a little nervous around the one-point compactification, as I recall there are traps awaiting the unwary when it comes to describing the functoriality. So I gave a more explicit description of at least one universal property (but wound up changing the text that was there). I apologize if the statement that had been there was correct (when suitably interpreted).

]]>[never mind]

]]>the entry *one-point compactification* did not use to point back to *vanishing at infinity*. I have now expanded the Idea-section with some brief remarks on this relation.

added to *one-point compactification* the remark that it is a functor on proper maps.

added also the following to *one-point compactification*, and cross-linked with *supension*:

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

$S^V \coloneqq (V)^\ast$for the $n$-sphere obtained as the one-point compactification of the vector space $V$.

+– {: .num_prop }

For $V,W \in Vect_{\mathbb{R}}$ two real vector spaces, there is a natural homeomorphism

$S^V \wedge S^W \simeq S^{V\oplus W}$between the smash product of their one-point compactifications and the one-point compactification of the direct sum.

=–

+– {: .num_remark }

In particular, it follows directly from this that the suspension $\Sigma(-) \simeq S^1 \wedge (-)$ of the $n$-spehere is the $(n+1)$-sphere, up to homeomorphism:

$\begin{aligned} \Sigma S^n & \simeq S^{\mathbb{R}^1} \wedge S^{\mathbb{R}^n} \\ & \simeq S^{\mathbb{R}^1 \oplus \mathbb{R}^n} \\ & \simeq S^{\mathbb{R}^{n+1}} \\ & \simeq S^{n+1} \end{aligned} \,.$=–

]]>added to *one-point compactification* a paragraph on an example:

]]>For $n \in \mathbb{N}$ the $n$-sphere (as a topological space) is the one-point compactification of the Cartesian space $\mathbb{R}^n$

$S^n \simeq (\mathbb{R}^n)^\ast$

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.

Okay, thanks. I had been worried about the previous definition of the open subsets. But maybe I just didn’t properly parse the sentence.

]]>I don’t think that anything in the original was incorrect. Instead, your version implies that the concept only makes sense for non-compact locally compact Hausdorff spaces.

But like the Stone–Čech compactification, it really makes sense for *any* topological space (or locale, in fact). If you reserve the term ‘compactification’ for open dense embeddings into compact Hausdorff spaces, then it is a compactification only in that case. But it exists regardless, as the original entry was supposed to make clear.

I have left the page as you designed it, but rewritten things to extend the definition again to any topological space (but not locales).

]]>tried to polish one-point compactification. I think in the process I actually corrected it, too. Please somebody have a close look.

]]>