# When is a bijective map between bundles a homeomorphism?

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
There should probably be some sort of commutative diagram. – Will Sawin Sep 26 '11 at 14:02
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
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

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