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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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What does $h$ have to do with it? –  Richard Kent Sep 26 '11 at 13:23
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There should probably be some sort of commutative diagram. –  Will Sawin Sep 26 '11 at 14:02
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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. –  Igor Belegradek Sep 26 '11 at 14:31
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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. –  Jack Huizenga Sep 26 '11 at 14:50
    
what is a heomeomorphism? –  euklid345 Sep 26 '11 at 21:10
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up vote 1 down vote accepted

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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And for vector bundles, there's a nice discussion in Milnor and Stasheff. –  Dan Ramras Sep 30 '11 at 6:58
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