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In my mind it is clear the formal definition of a fiber bundle but I can not have a geometric image of it. Roughly speaking, given three topological spaces $X, B, F$ with a continuous surjection $\pi: X\rightarrow B$, we "attach" to every point $b$ of $B$ a closed set $\pi^{-1}(b)$ such that it is homeomorphic to $F$ and so $X$ results a disjoint union of closed sets and each of them is homeomorphic to $F$. We also ask that this collection of closed subset of $X$ varies with continuity depending on $b\in B$, but I don't understand why this request is formalized using the conditions of local triviality.

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    A fiber bundle looks "locally" like a product. In some ways, it is to topological spaces what the "semi-direct product" is to groups.2012-10-22
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    In particular, it is more than just $\pi^{-1}(b)$ homeomorphic to $F$. For example, the disjoint union of copies of $F$, one for each $b\in B$, is not a fiber bundle, even though there is a function that satisfies $\pi^{-1}(b)\cong F$ which is continuous.2012-10-22
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    The (open) Möbius strip is a nice fiber bundle to use as an example. It is a nontrivial bundle whose base space is the circle and whose fiber is the real line. You make it from a strip of paper (think of it as $[0,1]\times\mathbb{R}$), which is a trivial bundle. Now identify $(0,x)$ with $(1,-x)$, and you have the Möbius strip. Understanding all the concepts for this simple example is a good start.2012-10-22
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    If I understand OP correctly, he is saying that he understands the definition but doesn't find it motivated by the geometry. His intuition is that the right geometric definition should be "all the fibers look alike, and they vary continuously," and so the issue with making the correct definition is figuring out how to formalize "they vary continuously." He doesn't see how that "the fibers vary continuously" is made formal by "looks locally like a product." He wants to be geometrically convinced that these are the same.2012-10-22
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    (continued) I tried to type something up, but don't really know what his picture is. Certainly it's clear that things which "look locally like products" have "continuously varying fibers." For the other direction, "pick a small enough neighborhood that the fibers are so close that you can bend them to look like a product."2012-10-22
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    One way to make this more precise is via the universal bundle. i.e. $\pi$ is isomorphic to the pullback of some "universal bundle" along a map $B \to Y$. In `reasonable circumstances' you can make $Y$ into the space of embeddings of $F$ into $\mathbb R^\infty$ modulo diffeomorphisms of $F$ (say if $F$ is smooth).2012-11-07

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