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nLab > Latest Changes: one-point compactification

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2010
    • (edited Nov 4th 2013)

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

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeOct 15th 2010

    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).

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2010

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

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2013

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

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

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

    Via this presentation of the nn-sphere the canonical action of the orthogonal group O(N)O(N) on ℝ n\mathbb{R}^n induces an action of O(n)O(n) on S nS^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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2013
    • (edited Nov 4th 2013)

    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≃(V) *S^n \simeq (V)^\ast. In this context and in view of the previous case, one usually writes

    S V≔(V) * S^V \coloneqq (V)^\ast

    for the nn-sphere obtained as the one-point compactification of the vector space VV.

    +– {: .num_prop }

    Proposition

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

    S V∧S W≃S V⊕W 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 Σ(−)≃S 1∧(−)\Sigma(-) \simeq S^1 \wedge (-) of the nn-spehere is the (n+1)(n+1)-sphere, up to homeomorphism:

    ΣS n ≃S ℝ 1∧S ℝ n ≃S ℝ 1⊕ℝ n ≃S ℝ n+1 ≃S n+1. \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} \,.

    =–

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 11th 2013

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

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2017

    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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2017
    • (edited May 15th 2017)

    [never mind]

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 15th 2017

    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).

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2017

    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

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2017

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

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2017

    Moreover, I have spelled out the proofs that:

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

    2. the inclusion X→X *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.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2017

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

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2017

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

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2019
    • (edited Sep 16th 2019)

    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:

    diff, v29, current

    • CommentRowNumber16.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 16th 2019

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

    • CommentRowNumber17.
    • CommentAuthorTim Campion
    • CommentTimeOct 15th 2019

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

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2019
    • (edited Oct 15th 2019)

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

    diff, v30, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2019
    • (edited Oct 21st 2019)

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

    diff, v31, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeDec 14th 2020

    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.

    diff, v38, current

    • CommentRowNumber21.
    • CommentAuthorSam Staton
    • CommentTimeDec 11th 2022

    Example of one-point compactification of N

    diff, v40, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeDec 11th 2022

    I have added some more hyperlinks to the technical terms in the example

    and have given it an Examples-environment (now here)

    in the course I slightly adjusted the section/example-title, please check if you agree.

    diff, v41, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeDec 11th 2022

    unrelated: while looking at this entry, I ended up fixing the pointer to Kelly’s book (here) as the previous link to archive.org/details/GeneralTopology no longer works

    diff, v41, current

    • CommentRowNumber24.
    • CommentAuthorSam Staton
    • CommentTimeDec 11th 2022

    Thanks, Urs! Added remark about subsequential spaces.

    diff, v42, current

    • CommentRowNumber25.
    • CommentAuthorGuest
    • CommentTimeApr 21st 2023

    I was reading this page and noticed an error in the universal property of one-point compactification (Section 3). As stated, the universal property essentially says that any continuous map f:X→Yf:X\to Y can be extended to a map f *:X *→Yf^*:X^*\to Y such that ∞↦y 0\infty\mapsto y_0 if f −1(K)f^{-1}(K) is compact whenever K⊆YK\subseteq Y is compact. I don’t think this is correct: consider, for instance, the identity map f:ℝ→ℝf:\mathbb{R}\to\mathbb{R}, which is a map between locally compact Hausdorff spaces such that the preimage of any compact set is compact. However, this clearly does not extend to a continuous map f:S 1→ℝf:S^1\to\mathbb{R}. A correct universal property (informally stated) is as follows: given topological spaces XX and YY, a continuous map f:X→Yf:X\to Y can be extended to a map f:X *→Yf:X^*\to Y such that ∞↦y 0\infty\mapsto y_0 if for any open neighborhood UU of y 0∈Uy_0\in U has f −1(Y∖U)f^{-1}(Y\setminus U) is compact. (I would change this myself but I’m unfamiliar with nLab’s editing practices and didn’t know if the mistake breaks something else in the article. Thanks for all your good work!)

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeApr 21st 2023
    • (edited Apr 21st 2023)

    Thanks for the heads-up.

    This statement originates in revision 20.

    I don’t think any other claim in the entry depends on it. Let’s fix it!

    But also, let’s add proof or reference.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2023
    • (edited Dec 5th 2023)

    added mentioning of the example (here) that the one-point extension of a compact space is the disjoint union with a point.

    (This deserves further discussion re the concept of compactification, but I leave it at that for the moment.)

    diff, v47, current

  28. added mentioning of the example (here) that the one-point extension of a compact space is the disjoint union with a point.

    (This deserves further discussion re the concept of compactification, but I leave it as that for the moment.)

    Not Anonymous

    diff, v48, current

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeFeb 14th 2024

    What “Not Anonymous” above actually did is fix a typo in “homeomorphic”.

    It seems to be a software bug that causes previous comments to be duplicated (because it keeps happening).

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