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Let $F \rightarrow E_i \rightarrow X_i$ be a bundle with fibre $F$ for i=1,2. Let $f:E_1 \rightarrow E_2$ be a bijective continuous map and $h: X_1 \rightarrow X_2$ a homeomorphism.

Is f then also a heomeomorphism? If not, what further properties are needed for f to be a hemeomorphism?

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    $\begingroup$ What does $h$ have to do with it? $\endgroup$
    – Autumn Kent
    Commented Sep 26, 2011 at 13:23
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    $\begingroup$ There should probably be some sort of commutative diagram. $\endgroup$
    – Will Sawin
    Commented Sep 26, 2011 at 14:02
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    $\begingroup$ Do you requre that $f$ descends to $h$? If so, the question makes sense, but the answer is obvious: $f$ need not be a homeomorphism, e.g. take $X_i$ to be a single point (or more generally, let the bundles be trivial), but choose $f$ so that its inverse discontinuous. $\endgroup$
    – Igor Belegradek
    Commented Sep 26, 2011 at 14:31
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    $\begingroup$ Take $F$ to be a space admitting a self-bijection that is not a homeomorphism, and take $X$ to be a one-point space. More generally, an obvious necessary condition for a general result is that every continuous bijection of $F$ be a homeomorphism. $\endgroup$
    – Jack Huizenga
    Commented Sep 26, 2011 at 14:50
  • $\begingroup$ what is a heomeomorphism? $\endgroup$
    – euklid345
    Commented Sep 26, 2011 at 21:10

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It has been pointed out in the comments that this sort of thing cannot hold for arbitrary fiber bundles.

To follow up on euklid345's comment regarding vector bundles, there is a statement of this type for arbitrary principal bundles, assuming the appropriate definitions. There's a detailed discussion of this in my course notes, available at http://sofia.nmsu.edu/~ramras/601.html (see p. 15 of Lectures 3-5). I believe it's also somewhere in Husemoller's book.

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  • $\begingroup$ And for vector bundles, there's a nice discussion in Milnor and Stasheff. $\endgroup$
    – Dan Ramras
    Commented Sep 30, 2011 at 6:58

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