Formalizing continuously indexed spaces in fiber bundles?

Question

This MSE question asks for clarification of the local triviality condition imposed in the definition of a fiber bundle.

As mentioned there, the point of local triviality seems to somehow ensure a "continuous variation of fibers". I would like to understand how this can be formalized (as suggested in the comments there).

Let $\pi:E\longrightarrow B$ denote our fiber bundle. The extension space $E$ is partitioned into homeomorphic copies of $F$ by its fibers: $$\left\{ \left\{b\right\}\times F \mid b\in B \right\}$$

We want this collection of sets to be continuously indexed by $B$. In section 2 of his Notes on Compactness, Martín Escardó makes the following definition:

Definition. Let $E$ be a topological space and $\left\{ V_b\mid b\in B \right\}$ be a family of subsets of $E$. This family will be called continuously indexed if the graph $\left\{ (x,b)\in E\times B\mid x\in V_b \right\}$ is open in the product topology in $E\times B$.

At any rate, this seems like exactly the kind of continuity condition we want from the partition corresponding to a fiber bundle, so I was hoping for something along the lines of

Partition by fibers is continuously indexed $(\implies,\iff,\impliedby$ bundle is locally trivial

And/or anything along those lines.

Finally, it would be nice if someone could explain where sheaves are related here, since they are supposed to represent continuously variable sets...

Answer 1

Suppose $F$ is homeomorphic to a compact subspace of $\Bbb R^\infty$. (There is some nice classification of which spaces this is true of, but I forget it. Maybe every compact Hausdorff space? At the very least it is true of any compact manifold and of any finite CW complex.)

Consider the space "$B(F)$" (terrible notation, sorry; this is meant to evoke "classifying space of $F$-bundles"), the elements of which are subsets of $\Bbb R^\infty$ which are homeomorphic to $F$ with the subspace topology, and equipped with the Hausdorff metric. This is the quotient of $\text{Emb}(F,\Bbb R^\infty)$ by the free etc action of $\text{Homeo}(F)$; and the first space is contractible, whence $B(F)$ is a $B\text{Homeo}(F)$. There is a tautological $F$-bundle over $B(F)$, given by the subset $E(F) \subset B(F) \times \Bbb R^\infty$, $E(F) = \{(S,x) | x \in S\}$. This is a locally trivial bundle.

This space does literally classify $F$-bundles over (paracompact) spaces $X$. But let's improve this a little bit. Let's say a 'concrete $F$-prebundle' (pre b/c not necessarily locally trivial) is a subset of $E \subset X \times E(F)$ such that the projection $E \to X$ has $F$ as every fiber. This is equivalent to saying that $E$ is the pullback of the tautological bundle over $B(F)$ by some map (not necessarily continuous!) $f: X \to B(F)$.

Then what you're looking for is probably as follows. TFAE. 1: $f$ is continuous; 2: the prebundle $E \to X$ is an honest $F$-bundle; 3: $E$ is a closed subset of $X \times E(F)$.

The proofs of these facts are straightforward and a little tedious; I will not endeavor to write them here, though I encourage someone else to and would gladly upvote such an answer. Note that the above only applies to compact fibers, which is why the example in my comment to the question does not work.