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

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