\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{schedule}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{schedules}{}\section*{{Schedules}}\label{schedules}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{schedules}{Schedules}\dotfill \pageref*{schedules} \linebreak
\noindent\hyperlink{paths}{Paths}\dotfill \pageref*{paths} \linebreak
\noindent\hyperlink{secschthm}{Schedule Theorem}\dotfill \pageref*{secschthm} \linebreak
\noindent\hyperlink{secglobal}{Globalisation Theorem}\dotfill \pageref*{secglobal} \linebreak
\noindent\hyperlink{secproof}{Proof of the Schedule Theorem}\dotfill \pageref*{secproof} \linebreak
\noindent\hyperlink{refs}{References}\dotfill \pageref*{refs} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Let $X$ be a [[topological space]] and $\mathcal{U}$ an [[open cover]] thereof. A ([[continuous function|continuous]]) path $\gamma \colon I \to X$ can pass through many of the elements of $\mathcal{U}$ as it winds its way around $X$. We can decompose that path into segments such that each segment lies wholly inside one of the [[open sets]] in $\mathcal{U}$. The \textbf{Schedule Theorem} says that this can be done continuously over all paths in $X$.

This was proved by \hyperlink{DyerEilenberg}{Dyer and Eilenberg} and applied to the question of fibrations over numerable spaces.

\hypertarget{schedules}{}\subsection*{{Schedules}}\label{schedules}

The idea of a schedule is that it is a way of decomposing a length into pieces and then assigning a label to each piece. This clearly fits with the stated purpose of these things since we wish to decompose a path into pieces and assign an open set to each piece.

To make this precise, we start with a set of labels. Following Dyer and Eilenberg, let us write this as $A$. Lengths are positive [[real numbers]] and so we also need the set of such, Dyer and Eilenberg denote this by $T$; thus $T \coloneqq \mathbb{R}_{\ge 0}$.

\begin{defn}
\label{schmon}\hypertarget{schmon}{}
The \textbf{schedule monoid} of $A$ is the [[free monoid]] on the [[set]] $A \times T$. It is written $S A$. Its elements are \textbf{schedules} in $A$.

\end{defn}
A schedule in $A$ is thus a finite [[order|ordered]] list of pairs $(a,t)$ where $a \in A$ and $t \in T$.

There are two notions of \emph{length} for a schedule. There is the \emph{word length} which simply counts the number of pairs. Then there is the function $l \colon S A \to T$ defined by $l((a_1,t_1) \cdots (a_k,t_k)) = t_1 + \cdots + t_k$. There is also a right action of $T$ on $S A$ which simply multiplies all of the lengths: $((a_1,t_1) \cdots (a_k,t_k)) \cdot t = ((a_1,t_1 t) \cdots (a_k,t_k t))$. Then $l(s t) = l(s) t$.

\begin{defn}
\label{reduced}\hypertarget{reduced}{}
A schedule is said to be \textbf{reduced} if all of its terms, $(a,t)$, have non-zero length, i.e. $t \gt 0$. The set of reduced schedules forms a submonoid of $S A$ which is written $R S A$.

\end{defn}
The empty schedule is reduced.

There is a [[retraction]] map $\rho \colon S A \to R S A$ defined by removing all terms with zero length part.

The schedule monoid is given a [[topology]] so that the labels are discrete and the lengths topologised as usual. More concretely, given a word $a_1 a_2 \dots a_k$ of elements in $A$, the set of schedules of the form $(a_1,t_1) (a_2,t_2) \cdots (a_k,t_k)$ is in bijection with $T^k$ and we make that bijection a homeomorphism. Then $S A$ is topologised by taking the coproduct over the set of words in $A$. The reduced schedule monoid is topologised as the quotient of this.

\hypertarget{paths}{}\subsection*{{Paths}}\label{paths}

Let $X$ be a [[topological space]]. Let $P X$ denotes its [[Moore path space]]. Suppose that we have a family $\mathcal{U}$ of subsets of $X$ indexed by some set $A$. Then we consider a schedule in $A$ as giving an ordered list of these subsets together with the times to be spent in each. For a path in $X$, and a schedule of the appropriate length, then we can ask whether or not the path \emph{fits} (or obeys) the schedule. We make that precise as follows.

\begin{defn}
\label{fits}\hypertarget{fits}{}
Suppose that we have $\alpha \in P X$ and $s \in S A$, and suppose that $s = (a_1, t_1) \cdots (a_k,t_k)$. Then we say that $\alpha$ \textbf{fits the schedule} s, written $\alpha \Vert s$, if the following conditions hold:

\begin{enumerate}%
\item $l(\alpha) = l(s)$
\item We can split $\alpha$ into subpaths according to the times $\{t_i\}$. Let $\alpha_i$ be the $i$th segment. Then $\alpha_i \in P U_{a_i}$.

\end{enumerate}
\end{defn}
Here, $l \colon P X \to T$ is the function that assigns to a Moore path its length. The schedule designates a decomposition of $[0,l]$ into subintervals with $t_i$ being the length of the $i$th subinterval. Then saying that $\alpha$ fits the schedule $s$ means that $\alpha$ spends the $i$th subinterval in the open set $U_{a_i}$.

\hypertarget{secschthm}{}\subsection*{{Schedule Theorem}}\label{secschthm}

We can now state the main theorem.

\begin{theorem}
\label{schthm}\hypertarget{schthm}{}
Let $X$ be a [[topological space]]. Let $\mathcal{U}$ be a [[locally finite cover|locally finite]] [[open covering]] of $X$ by [[numerable]] open sets with indexing set $A$. Then there is a covering $\mathcal{F}$ of $P X$ by [[closed sets]] and a family of [[continuous functions]] $f \colon F \to S A$, indexed by $F \in \mathcal{F}$ such that:

\begin{enumerate}%
\item for each $\alpha \in P X$, there some finite subfamily $\{F_1, \dots, F_k\} \subseteq \mathcal{F}$ such that $\alpha$ is in the interior of $\bigcup F_j$,
\item for each $\alpha \in F$, $\alpha \Vert f_F(\alpha)$, and
\item for each $\alpha \in F \cap F'$, $\rho(f_F(\alpha)) = \rho(f_{F'}(\alpha))$

\end{enumerate}
\end{theorem}
The first condition is purely about the covering. Dyer and Eilenberg use the term \emph{local covering} for a covering by closed sets with this property.

\begin{corollary}
\label{schcor}\hypertarget{schcor}{}
There exists a [[continuous function]] $h \colon P X \to R S A$ such that $\alpha \Vert h(\alpha)$ if $l(\alpha) \gt 0$ and $h(\alpha) = \Lambda$ if $l(\alpha) = 0$.

\end{corollary}
Here, $\Lambda \in R S A$ is the empty word.

\hypertarget{secglobal}{}\subsection*{{Globalisation Theorem}}\label{secglobal}

The original motivation for the notion of \emph{schedules} was to prove the \emph{globalisation theorem} for (Hurewicz) [[Hurewicz fibration|fibrations]].

\begin{theorem}
\label{global}\hypertarget{global}{}
Let $p \colon Y \to B$ be a continuous function. Suppose that $\mathcal{U}$ is a locally finite covering of $B$ by numerable open sets with the property that for each $U \in \mathcal{U}$ then the restriction $p_U \colon Y_U \to U$ is a fibration. Then $p$ is a fibration.

\end{theorem}
The link between the globalisation theorem and the schedule theorem is the characterisation of Hurewicz fibrations in terms of [[Hurewicz connections]].

\hypertarget{secproof}{}\subsection*{{Proof of the Schedule Theorem}}\label{secproof}

Let $X$ be a topological space. Let $\mathcal{U}$ be a locally finite open covering of $X$ by numerable open sets and indexing set $A$.

Let us write $A^*$ for the free monoid on $A$. Then there is a function $A^* \times T \to S A$ which takes $(a_1 a_2 \cdots a_k, t)$ to the schedule $(a_1,t/k)(a_2,t/k)\cdots (a_k,t/k)$. We say that a path $\alpha \in P X$ \emph{evenly fits} $s \in A^*$, and write this as $\alpha \Vert_e s$, if it fits the schedule corresponding to $(s,l(\alpha))$.

We need an initial technical result.

\begin{lemma}
\label{even}\hypertarget{even}{}
There is a locally finite covering $\mathcal{W} = \{W_s \mid s \in A^*\}$ of $P X$ by numerable open sets such that for $\alpha \in W_s$ then $\alpha$ evenly fits the word $s$.

\end{lemma}
\begin{uremark}
Let us explain why this is a reasonable result. Consider a path, $\alpha$, of length $l$. We pull back the cover $\mathcal{U}$ to a cover of $[0,l]$. Using compactness of $[0,l]$ we can replace the pull-back cover by a finite family of open subintervals of $[0,l]$ which cover $[0,l]$. Each subinterval is labelled by an element of $\mathcal{U}$ (though a label might be reused). As the family is finite, the intersections are finite and therefore have a minimum length. Choose $n$ big enough so that $l/n$ is less than this minimum length. Then consider the subdivision of $[0,l]$ given by $\{0,1/n,\dots,l/n\}$. Our conditions on $n$ guarantee that every intersection of subintervals contains at least one of these division points. We can therefore assign to each subinterval of the form $[k/n, (k+1)/n]$ one of the original family of subintervals that contains it. Then we can assign the corresponding element of $\mathcal{U}$. Thus $\alpha$ fits evenly the corresponding word.

Thus the sets $Y_s \coloneqq \{\alpha : \alpha \|_e s\}$ cover $P X$. That they are open follows from the fact that the condition for membership depends on certain compact sets lying in certain open sets and we use the [[compact-open topology]] on $P X$.

What is more complicated is reducing the family to a locally finite one.

\end{uremark}
As $\mathcal{W}$ is locally finite and its elements are numerable, we can choose a numeration that is also a partition of unity. That is, we can choose continuous functions $q_s \colon X \to [0,1]$ with the property that $q_s^{-1}((0,1]) = W_s$ and $\sum_s q_s = 1$.

Let $\mathcal{B}$ be the set of finite subsets of $A^* \setminus \Lambda$ (where $\Lambda$ is the empty word). For $b \in \mathcal{B}$ we define

\begin{displaymath}
\begin{aligned}
D_b &\coloneqq \{\alpha \in P X \mid \sum_{s \in b} q_s(\alpha) = 1 \} \\
&=\{ \alpha \in P X \mid q_s(\alpha) = 0 \; \text{for all}\; s \notin b\}
\end{aligned}
\end{displaymath}
This is a covering of $P X$ by closed sets. As $\mathcal{W}$ is locally finite, for $\alpha \in P X$ there is some neighbourhood $V$ which meets only a finite number of the $\mathcal{W}$. These are indexed by elements of $A^*$, indeed of $A^* \setminus \Lambda$, and so the set of indices is an element, say $b$, of$\mathcal{B}$. Then for $s \notin b$, $q_s \mid V = 0$ and so for $\beta \in V$, $\sum_{s \in b} q_s(\beta) = 1$, whence $V \subseteq D_b$. Thus each $\alpha$ is contained in the interior of some $D_b$.

Now let us put a [[total ordering]] on $A^*$. This induces a total ordering on each $b \in \mathcal{B}$ and thus allows us to define the partial sums of the summation $\sum_{s \in b} q_s$. Write these as $Q_i$, with $Q_0$ as the zero function.

Fix $b \in \mathcal{B}$ and write it as $b = \{s_1,s_2,\dots,s_k\}$ in the inherited ordering. Let $e = (l_1,r_1,\dots,l_k,r_k)$ be a list of integers with the property that $1 \le l_i \le r_i \le \#s_i$ where $\#s_i$ is the word length of $s_i$. Define:

\begin{displaymath}
D_{(b,e)} = \left\{ \alpha \in D_b \mid \frac{l_i -1}{\# s_i} \le Q_{i - 1}(\alpha) \le \frac{l_i}{\# s_i} \; \text{and} \; \frac{r_i - 1}{\# s_i} \le Q_i(\alpha) \le \frac{r_i}{\# s_i} \right\}.
\end{displaymath}
This is closed in $D_b$ and the collection $\{D_{(b,e)}\}$ is a finite cover of $D_b$. The family $\{D_{(b,e)}\}$ ranging over all $b \in \mathcal{B}$ and suitable $e$ is the family $\mathcal{F}$ that we are looking for. It has the required covering property since the interiors of the $D_b$ cover $P X$.

Define $f_{(b,e)} \colon D_{(b,e)} \to S A$ as follows:

\begin{displaymath}
f_{(b,e)}(\alpha) = \sigma_1 \cdots \sigma_k l(\alpha)
\end{displaymath}
where $\sigma_i$ is the schedule with $\# \sigma_i = r_i - l_i + 1$ and $l(\sigma_i) = q_{s_i}(\alpha)$, and if $s_i = a_1 \cdots a_n$ then if $l_i \lt r_i$ we have

\begin{displaymath}
\sigma_i = \left( a_{l_1}, \frac{l_i}{n} - Q_{i - 1}(\alpha)\right) \left(a_{l_i+1}, \frac{1}{n} \right) \cdots \left(a_{r_i - 1}, \frac{1}{n} \right) \left( a_{r_i}, Q_i(\alpha) - \frac{r_i - 1}{n} \right)
\end{displaymath}
otherwise, $\sigma_i = (a_{l_i}, Q_i(\alpha) - Q_{i-1}(\alpha))$.

This is continuous and for $\alpha \in D_{(b,e)}$ then $\alpha$ fits $f_{(b,e)}(\alpha)$. Moreover, for $\alpha \in D_{(b,e)} \cap D_{(b',e')}$ then $\rho f_{(b,e)}(\alpha) = \rho f_{(b',e')}(\alpha)$.

\hypertarget{refs}{}\subsection*{{References}}\label{refs}

\begin{itemize}%
\item Dyer, E. and E., Samuel. (1988). Globalizing fibrations by schedules. \emph{Fund. Math.}, \emph{130}, 125--136. \href{http://www.ams.org/mathscinet-getitem?mr=963792}{MR0963792}


\item Dyer, Eilenberg, MR0963792



\end{itemize}
[[!redirects schedules]]

[[!redirects schedule theorem]]



\end{document}