# Formalizing continuously indexed spaces in fiber bundles?

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...

• Your statement is almost certainly false. Consider $B = \Bbb R$, $V_b = \Bbb Z$ for $b \geq 1$, $\Bbb Z$ for $b \leq 0$, and in between, $\Bbb N \cup (b+-\Bbb N)$. This is not locally trivial but is continuously indexed. – user98602 May 23 '16 at 14:52
• @MikeMiller what is $\Bbb N \cup (b+-\Bbb N)$? At any rate, I'm looking for connections and implications more than anything. I just don't really understand why being locally a product projection means continuously varying fibers. For instance, why do fibers vary continuously for trivial fiber bundles? – Arrow May 23 '16 at 15:00
• The definition of "continuously indexed" in the question seems inappropriate for the use you want to make of it. It implies that each of the sets $V_b$ has to be open, which is the case in Escardó's situation but not in yours. – Andreas Blass May 23 '16 at 18:10
• Sorry, yes, my comment does not apply to the sense of continuously indexed in the question. I just mean that they depend continuously on $b$ in Hausdorff metric which is the most reasonable candidate I can think of for the notion of 'continuously indexed'. – user98602 May 23 '16 at 18:39

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 $X \times E(F)$.