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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 24th 2017
    • (edited May 24th 2017)

    at vector bundle I have spelled out the proof that for XX paracompact Hausdorff then the restrictions of vector bundles over X×[0,1]X \times [0,1] to X×{0}X \times \{0\} and X×{1}X \times \{1\} are isomorphic.

    It’s just following Hatcher, but I wanted to give full detail to the argument of what is now this lemma.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 24th 2017

    You didn’t want to give the argument for numerable bundles on arbitrary spaces? The proof is exactly the same.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 24th 2017

    Sorry, which statement are your referring to? The only place where I assumed something extra, namely paracompact Hausdorff, is this prop. For the proof of that I seem to need a partition of unity, no?

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 24th 2017
    • (edited May 24th 2017)

    The statement in #1 generalises from arbitrary vector bundles on paracompact spaces to numerable bundles on arbitrary spaces. The proof uses the so-called stacking lemma, which is in Dold’s Algebraic Topology, section A.2. Numerable bundles are those that trivialise over an open cover with a subordinate partition of unity (so all bundles when on a paracompact space).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2017

    I see. So I was headed for the discussion of the classifying space, where I need all bundles. But feel invited to add this remark.

    If you do so, notice that I am splitting off the material on toopological vector bundles from the main entry “vector bundle” to topological vector bundle. More on this in the next comment.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2017
    • (edited May 25th 2017)

    I have expanded the Idea-section at vector bundle a fair bit.

    Then, in view of the recent disussion with Todd, I am splitting off an entry topological vector bunde for discussion of the standard topological stuff (no sheaf semantics etc.).

    I moved over the corresponding material on Definition and properties. Then I polished the definition material at topological vector bundle, or at least two thirds of it. Need to interrupt for a moment.

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