nForum - Discussion Feed (one-point compactification) 2019-07-22T18:25:50-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Urs comments on "one-point compactification" (63005) https://nforum.ncatlab.org/discussion/1968/?Focus=63005#Comment_63005 2017-05-31T03:54:30-04:00 2019-07-22T18:25:49-04:00 Urs https://nforum.ncatlab.org/account/4/ I have added the remark here that every compact Hausdorff space is the one-point compactification of its complement by one point.

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

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Urs comments on "one-point compactification" (62814) https://nforum.ncatlab.org/discussion/1968/?Focus=62814#Comment_62814 2017-05-23T09:01:01-04:00 2019-07-22T18:25:49-04:00 Urs https://nforum.ncatlab.org/account/4/ I have made exlicit the proof that the one-point compactification of &Ropf; n\mathbb{R}^n is S nS^n (here)

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

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Urs comments on "one-point compactification" (62685) https://nforum.ncatlab.org/discussion/1968/?Focus=62685#Comment_62685 2017-05-15T13:56:25-04:00 2019-07-22T18:25:49-04:00 Urs https://nforum.ncatlab.org/account/4/ Moreover, I have spelled out the proofs that: if XX is locally compact then X &ast;X^\ast is Hausdorff precisely if XX is (here) the inclusion X&rightarrow;X &ast;X \to X^\ast is an ...

Moreover, I have spelled out the proofs that:

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

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

3. 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.

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Urs comments on "one-point compactification" (62683) https://nforum.ncatlab.org/discussion/1968/?Focus=62683#Comment_62683 2017-05-15T13:12:40-04:00 2019-07-22T18:25:50-04:00 Urs https://nforum.ncatlab.org/account/4/ just to be more boring still, I made explicit why X &ast;X^\ast is indeed compact: here

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

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Urs comments on "one-point compactification" (62682) https://nforum.ncatlab.org/discussion/1968/?Focus=62682#Comment_62682 2017-05-15T13:01:09-04:00 2019-07-22T18:25:50-04:00 Urs https://nforum.ncatlab.org/account/4/ 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 bore everyone, I have written out in detail why the topology on the one-point extension is well defined in the first place: here

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Todd_Trimble comments on "one-point compactification" (62680) https://nforum.ncatlab.org/discussion/1968/?Focus=62680#Comment_62680 2017-05-15T12:20:56-04:00 2019-07-22T18:25:50-04:00 Todd_Trimble https://nforum.ncatlab.org/account/24/ 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 ...

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).

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Urs comments on "one-point compactification" (62679) https://nforum.ncatlab.org/discussion/1968/?Focus=62679#Comment_62679 2017-05-15T11:27:12-04:00 2019-07-22T18:25:50-04:00 Urs https://nforum.ncatlab.org/account/4/ [never mind]

[never mind]

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Urs comments on "one-point compactification" (60925) https://nforum.ncatlab.org/discussion/1968/?Focus=60925#Comment_60925 2017-01-14T04:50:37-05:00 2019-07-22T18:25:50-04:00 Urs https://nforum.ncatlab.org/account/4/ 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.

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.

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Urs comments on "one-point compactification" (43089) https://nforum.ncatlab.org/discussion/1968/?Focus=43089#Comment_43089 2013-11-11T11:23:12-05:00 2019-07-22T18:25:50-04:00 Urs https://nforum.ncatlab.org/account/4/ added to one-point compactification the remark that it is a functor on proper maps.

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

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Urs comments on "one-point compactification" (42907) https://nforum.ncatlab.org/discussion/1968/?Focus=42907#Comment_42907 2013-11-04T08:11:02-05:00 2019-07-22T18:25:50-04:00 Urs https://nforum.ncatlab.org/account/4/ added also the following to one-point compactification, and cross-linked with supension: Slightly more generally, for VV any real vector space of dimension nn one has S n&simeq;(V) ...

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 }

###### Proposition

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 }

###### 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} \,.

=–

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Urs comments on "one-point compactification" (42904) https://nforum.ncatlab.org/discussion/1968/?Focus=42904#Comment_42904 2013-11-04T06:58:18-05:00 2019-07-22T18:25:50-04:00 Urs https://nforum.ncatlab.org/account/4/ added to one-point compactification a paragraph on an example: For n&Element;&Nopf;n \in \mathbb{N} the nn-sphere (as a topological space) is the one-point compactification of the ...

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.

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Urs comments on "one-point compactification" (17207) https://nforum.ncatlab.org/discussion/1968/?Focus=17207#Comment_17207 2010-10-15T19:15:46-04:00 2019-07-22T18:25:50-04:00 Urs https://nforum.ncatlab.org/account/4/ Okay, thanks. I had been worried about the previous definition of the open subsets. But maybe I just didn’t properly parse the sentence.

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

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TobyBartels comments on "one-point compactification" (17200) https://nforum.ncatlab.org/discussion/1968/?Focus=17200#Comment_17200 2010-10-15T17:59:14-04:00 2019-07-22T18:25:50-04:00 TobyBartels https://nforum.ncatlab.org/account/7/ 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 ...

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).

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Urs comments on "one-point compactification" (17183) https://nforum.ncatlab.org/discussion/1968/?Focus=17183#Comment_17183 2010-10-15T09:09:31-04:00 2019-07-22T18:25:50-04:00 Urs https://nforum.ncatlab.org/account/4/ tried to polish one-point compactification. I think in the process I actually corrected it, too. Please somebody have a close look.

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

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