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

    S n( 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 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,WVect V,W \in Vect_{\mathbb{R}} two real vector spaces, there is a natural homeomorphism

    S VS WS VW 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 1S n S 1 n S 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 XX *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 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.