Locally trivial fiber bundle

A locally trivial fiber bundle with fiber $F$ is a continuous map $p:E\to B$, an open cover $(U_i{)}_{i\in S}$ of $B$, and homeomorphisms ${\theta }_i:p^{-1}(U_i)⟶U_i\times F,{\mathrm{pr}}_1{\theta }_i=p{\mathrm{\mid }}_{p^{-1}(U_i)}\mathrm{.}$ Here bundle charts, including those used later for principal and associated bundles, are charts in ordinary topological spaces with ordinary product and subspace topologies. A bundle whose spaces are CGWH in addition can be considered in the CGWH fibration convention; we do not silently replace an ordinary product chart by a weaker k-product-chart assumption. Products and homeomorphisms have the meanings of A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice and Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological.

On $U_i\cap U_j$, the coordinate change has the form $(b,x)\mapsto (b,g_{ji}(b,x))$, with continuous inverse $(b,y)\mapsto (b,g_{ij}(b,y))$. In particular each $g_{ji}(b,-)$ is a homeomorphism of $F$. The identities $g_{ii}(b,x)=x$ and $g_{ki}(b,x)=g_{kj}(b,g_{ji}(b,x))$ hold because the corresponding chart composites cancel. No topology on a homeomorphism group of $F$ is assumed.

The bundle is numerable when the data include a partition $({\rho }_i:B\to [0,1]{)}_{i\in S}$ with locally finite cozero sets, sum one, and $\mathrm{supp}({\rho }_i)=\overline{\{b:{\rho }_i(b)>0\}}\subseteq U_i\mathrm{.}$ This is the support-subordinate convention of Locally finite partitions of unity and subordination to an open cover. Repeating a chart with multiple indices is allowed, so a partition subordinate to a refinement can be accompanied by an explicitly assigned original chart. Merely having pointwise finite sums or cozero containment is not this specified numerating data.

An empty base forces $E$ empty and allows the empty cover and partition. The fiber $F$ may be empty: then every chart domain, hence $E$, is empty even when $B$ is nonempty. When $F$ is nonempty the charts imply surjectivity by taking one point in the chart fiber for each particular base point; this assertion does not require choosing a simultaneous section. Over a point a chart identifies $E$ with $F$. No AC is assumed in the definition.