I was reading "A Concise Course in Algebraic Topology" by J.P.May (page 52) and found the proof of the following theorem incomprehensible:


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


The only if part is clear. To check the if part, May tries to patch together path lifting functions of $p|p^{-1} (U)$ to construct a path lifting function. To do this, he defines some complicated functions, which I could not easily understand.


He starts by choosing maps $\lambda\_U : B\rightarrow I$ such that $\lambda \_U ^{-1} (0,1]=U$. For each finite ordered subset $T=\{U\_1 , \cdots , U\_n \}$ of $\mathcal O$, he defines $c(T)=n$, and $\lambda \_T : B^I \rightarrow I$ as $$\lambda\_T (\beta ) = \inf \{(\lambda\_{U\_i} \circ \beta ) (t) : (i-1)/n \le t \le i/n , 1\le i\le n \}$$
and let $W\_T = \lambda \_T ^{-1} (0,1]$. I understand that $\{W\_T\}$ is an open cover of $B^I$, and that $\{W\_T : c(T)<n\}$ is locally finite for each $n$. Then he defines $\lambda \_T : B^I \rightarrow I $ as 
$$\lambda\_T (B) = \max \{ 0,\lambda\_T (B) - n \sum \_{c(S)<c(T)} \lambda\_S (\beta )\}, V\_T = \{\beta : \gamma\_T (\beta)>0\} \subset W\_T.$$
I understand that $\{V\_T\}$ is a locally finite open cover of $B^I$.


Now, he chooses path lifting functions $$s\_U : p^{-1} (U) \times \_p U^I \rightarrow p^{-1} (U) ^I $$ for each $U\in \mathcal O $, and he constructs a global path lifting function by the following steps.


First, choose a total order on the sets of finite ordered subsets of $\mathcal O $. For a given $T=\{U\_1, \cdots , U\_n \}$, he defines $s\_T (e,\beta [u,v]):[u,v]\rightarrow E $ for each $\beta \in V\_T$, $u,v\in [0,1]$, $e\in p^{-1} (\beta (u))$ canonically, by lifting $\beta[u,v]$ by letting its starting point be $e$. ($\beta [u,v]$ is defined as the restriction of $\beta$ to $[u,v]$.)


Now here is the part where I don't understand at all. For each $e,\beta$ such that $\beta(0)=e$, he defines $s(e,\beta)$ to be the concatenation of the parts $s\_{T\_j} (e\_{j-1} , \beta [u\_{j-1} ,u\_j ])$, $1\le j \le q$ where the $T\_i $ run through the set of all $T$ such that $\beta \in V\_T$, and $$u\_0 =0, \ u\_j = \sum \_{i=1} ^j \gamma \_{T\_i } (\beta ).$$ 


He asserts that $s$ is well defined and continuous, thus a path lifting function for $p$.


I could not understand at all why the definition of $s$ makes sense. For this to make sense, I believe that $u\_q$ should be $1$, which I cannot get at all. For me, this definition seems completely random. Can someone please kindly explain why this definition makes sense?



Proof that a local fibration is a fibration, in May
May has carelessly made a typo in transcribing a proof in one of his books to another,
and it has taken an impressive high school student in Korea to catch it. In his earlier book ``Classifying 
Spaces and Fibrations" --- the title is a joke: "classifying" here has two meanings ---
(<http://www.math.uchicago.edu/~may/BOOKS/Classifying.pdf>)
there is a generalization of the cited result in "Concise". In its proof, but using
the notation of "Concise", May remembers to normalize by setting 
$u\_j = \sum\_{i=1}^{j}\gamma\_{T\_i}(\beta) / \sum\_{i=1}^{q}\gamma\_{T\_i}(\beta).$
With this correction (and correction of another typo, $s(e,\beta)= e$ should read
$s(e,\beta)(0) = e$) the proof is correct, albeit miserably hard to read, way too concise. May should apologize to his readers.


