Evaluation is a numerable bundle and Hurewicz fibration

Local product charts. Let $\xi \in B$ and let $U_{\xi }$ and $s_{\xi }$ be the neighbourhood and continuous section provided by [L1]; the evaluation map is continuous because for $h,h^{\mathrm{\prime }}\in E$ one has $\parallel h(q_i)-h^{\mathrm{\prime }}(q_i){\parallel }_2\le d(h,h^{\mathrm{\prime }})$ for all $i$ and the quotient map $F_n(\mathrm{int}{D}^2)\to B$ of [L7] is continuous. Define ${\theta }_{\xi }:{\mathrm{ev}}^{-1}(U_{\xi })⟶U_{\xi }\times F,{\theta }_{\xi }(h):=(\mathrm{ev}(h),s_{\xi }(\mathrm{ev}(h){)}^{-1}\circ h)\mathrm{.}$ For $h\in {\mathrm{ev}}^{-1}(U_{\xi })$ the composite $s_{\xi }(\mathrm{ev}(h){)}^{-1}\circ h$ lies in $F$: applying it to the set $Q_n$ gives $s_{\xi }(\mathrm{ev}(h){)}^{-1}(\mathrm{ev}(h))=Q_n$ because $s_{\xi }(\mathrm{ev}(h))(Q_n)=\mathrm{ev}(h)$ as sets. The map ${\theta }_{\xi }$ is continuous, being built from $\mathrm{ev}$, the continuous section, inversion and composition, which are continuous by [L8]; it satisfies ${\mathrm{pr}}_1\circ {\theta }_{\xi }=\mathrm{ev}$; and it is a bijection with inverse $(\xi ,g)\mapsto s_{\xi }(\xi )\circ g$, because $\mathrm{ev}(s_{\xi }(\xi )\circ g)=[s_{\xi }(\xi )(g(Q_n))]=[s_{\xi }(\xi )(Q_n)]=\xi$ and $s_{\xi }(\xi {)}^{-1}\circ (s_{\xi }(\xi )\circ g)=g$, while the other composite is the identity by the same computation. Hence the maps ${\theta }_{\xi }$ are local product charts over the open cover $\{U_{\xi }\}$ and $\mathrm{ev}$ is a locally trivial fiber bundle with fiber $F$ as in [L2]; the fibre over $[Q_n]$ is exactly $F$ by [L8].

Numerating data. For $n\ge 1$, give the ordered configuration space the metric $d(x,y)={\max }_i\parallel x_i-y_i{\parallel }_2$ and set $d_B([x],[y]):={\min }_{\sigma \in S_n}d(x,\sigma y)\mathrm{.}$ Coordinate permutations are isometries, so this is independent of representatives and symmetric. The finite minimum is zero exactly for equal orbits; composing minimizing permutations and applying the triangle inequality for $d$ gives the triangle inequality for $d_B$. Moreover the preimage of the $d_B$-ball about $[x]$ of radius $r$ is the union of the permutation translates of the ordered $r$-ball about $x$, hence is open. Conversely, the preimage of a quotient-open neighbourhood of $[x]$ contains an ordered ball about $x$ and is permutation-invariant, so it contains that union. Thus $d_B$ gives exactly the quotient topology of [L7]. For $n=0$, $B$ is a singleton and is metrizable. The family $\{U_{\xi }\}$ is therefore an open cover of the metric space $B$. By [L6] choose one such pair $(U_{\xi },s_{\xi })$ for each $\xi \in B$; using AC we also obtain DC and countable choice by [L4], so [L3] supplies a locally finite partition of unity $({\rho }_{\xi })$ subordinate to the cover with all sums equal to one. Since each support satisfies $\mathrm{supp}{\rho }_{\xi }\subseteq U_{\xi }$, the charts of step 1.1 together with this partition are exactly the numerating data required in [L2]; hence the bundle is numerable.

The fibration. By [L5] and AC the numerable bundle just produced, with its displayed charts and partition of unity, is a Hurewicz fibration. The same argument applies for $n=0$, where $B$ is a one-point space: the unique chart identifies $E$ with the fibre, the constant partition with value one is locally finite, and the bundle over a point is a Hurewicz fibration.