Details in May's proof that local fibrations are fibrations J. P. May proves on pages 49–50 (51–52 in the online pdf) of A Concise Course in Algebraic Topology that local (Hurewicz) fibrations are (Hurewicz) fibrations: 

Theorem. Let $p: E \to B$ be a map and let $\mathscr{O}$ be a numerable open cover of $B$. 
  Then $p$ is a fibration if and only if $p: p^{-1}(U) \to U$ is a fibration for every $U \in \mathscr{O}$.

(Note: He defines a numerable open cover as a locally finite open cover together with continuous maps $\lambda_U: B \to I$ such that $\lambda_U^{-1}(0,1] = U$.)
To do this, he first sets up some scaffolding for a local-to-global argument. 
Using the numerability of $\mathscr{O}$, he chooses maps $\lambda_U: B \to I$ with $\lambda_U^{-1}(0,1] = U$. 
Given an ordered subset $T = \{U_1, \dots, U_n\}$ of sets in $\mathscr{O}$ (that I believe need not be distinct, please correct me if I'm wrong), he then defines $c(T) = n$ and $\lambda_T: B^I \to I$ by 
$$\lambda_T(\beta) = \inf\{(\lambda_{U_i} \circ \beta)(t) \mid (i-1)/n \leq t \leq i/n, 1 \leq i \leq n\}.$$
He proceeds to define $W_T := \lambda_T^{-1}(0,1] =  \{\beta \mid \beta(t) \in U_i \text{ if } t \in[(i-1)/n, i/n]\} \subset B^I$, 
so that $\{W_T\}$ is an open cover of $B^I$. 
He states that $\{W_T \mid c(T)<n\}$ is locally finite for each fixed $n$, 
which follows from the local finiteness of $\mathscr{O}$. 
Now, if $c(T) = n$, he defines $\gamma_T: B^I \to I$ by
$$\gamma_T(\beta) = \max\{0, \lambda_T(\beta) - n{\textstyle\sum}_{c(S)<n}\lambda_S(\beta)\}$$
and $V_T := \{\beta \mid \gamma_T(\beta) > 0\}=\{ \beta \mid \lambda_T(\beta) > n{\textstyle\sum}_{c(S)<n}\lambda_S(\beta)\} \subset W_T$. He claims that $\{V_T\}$ is a locally finite open cover of $B^I$. 
My questions, then, are the following:


*

*How do I show that $\lambda_T$ is continuous? To prove that the sets $W_T = \lambda_T^{-1}(0,1]$ are open, I just resorted to showing that it was the intersection $\bigcap_{1 \leq i \leq n} \{\beta \mid \beta([(i-1)/n,i/n]) \subset U_i\}$ of subbase sets of $B^I$ as it is equipped with the ($k$-ification of the) compact-open topology. But it's probably better to use the continuity of $\lambda_T$ (as it would also follow easily that the sets $V_T$ are open).

*Why is $V_T$ defined so, and why doesn't the rest of the patching argument that follows in May's book work if I use $W_T$ and $\lambda_{T_i}$ in place of $V_T$ and $\gamma_{T_i}$? (Note that there's a typo in the last part of the proof in May's book: it should be $u_j = \sum_{i=1}^j\gamma_{T_i}(\beta)/\sum_{i=1}^q\gamma_{T_i}(\beta)$, as pointed out in this MathOverflow post.)

*How do I show that $\{V_T\}$ is a cover of $B^I$? I have no idea on this one—for $\{W_T\}$, I was able to use the Lebesgue number lemma to guarantee that $\lambda_T(\beta)>0$ for a given curve $\beta$, but it's not clear to me why there should exist $T$ such that $\lambda_T(\beta)>n\sum_{c(S)<n}\lambda_S(\beta)$—this ties in with my second question above. If I can show that $\{V_T\}$ a cover of $B^I$, local finiteness is then inherited from $\{W_T\}.$
Thanks!
 A: I found the answers to questions 2 and 3 in E. H. Spanier's Algebraic Topology (Theorem 2.7.12, pp. 96):


*$\{V_T\}$ is essentially $\{W_T\}$, but locally finite—this is crucial for making sure the sums are finite and well-behaved.

*Let $\beta \in B^I$, and let $m$ be the smallest integer such that $\lambda_T(\beta)>0$ for some $T$ with $c(T)=m$. Then $\sum_{c(S)<m}\lambda_S(\beta)=0$, so that $\gamma_T(\beta)>0$ as desired. I also noticed from Spanier that it takes some extra work for local finiteness to then be inherited from $\{W_T\}$. Indeed, assume $N$ is chosen such that $N>m$ and $\lambda_T(\beta)>1/N$. Then $\sum_{c(S)<N}\lambda_S(\beta)>1/N$ (as $\lambda_T(\beta)$ itself is included in the sum); in particular, $N\sum_{c(S)<N}\lambda_S(\beta')>1$ for all $\beta'$ in some neighborhood $V$ of $\beta$ by continuity, so that all functions $\gamma_T$ with $c(T) \geq N$ vanish on $V$. It follows that the corresponding set $V_T$ is disjoint from $V$, so that the local finiteness of $\{V_T\}$ (at $\beta$) follows from that of $\{W_T \mid c(T)<N\}$.
