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tried to polish one-point compactification. I think in the process I actually corrected it, too. Please somebody have a close look.
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).
Okay, thanks. I had been worried about the previous definition of the open subsets. But maybe I just didn’t properly parse the sentence.
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.
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 the remark that it is a functor on proper maps.
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.
[never mind]
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).
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
just to be more boring still, I made explicit why $X^\ast$ is indeed compact: 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.
I have made exlicit the proof that the one-point compactification of $\mathbb{R}^n$ is $S^n$ (here)
I have added the remark here that every compact Hausdorff space is the one-point compactification of its complement by one point.
added pointer to what I guess is the original reference:
and equipped the pointer to Kelly’s textbook with hyperlinks to the arXiv scan copy:
Internet Archive scan, not arXiv. I don’t think the arXiv would like giant scan of a published book uploaded!
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!
expanded (here) the subsection previous titled “Functoriality”, now “Monoidal functoriality”:
Added a reference for the statement of functoriality with respect to proper maps, and added the statement that one-point compactification intertwines Cartesian product with smash product.
Further down we used to have this statement for (just) representation spheres, and I have added cross-links now.
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