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\begin{document}

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\section*{numerable open cover}

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

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

[[!include topology - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_a_coverage_as_a_site}{As a coverage, as a site}\dotfill \pageref*{as_a_coverage_as_a_site} \linebreak
\noindent\hyperlink{relation_to_numerable_bundles}{Relation to numerable bundles}\dotfill \pageref*{relation_to_numerable_bundles} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{numerable open cover} is an [[open cover]] of a [[topological space]] that admits a subordinate [[partition of unity]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $\{U_i\}$ be an open [[cover]] of the [[topological space]] $X$ (actually Dold doesn't always require open, see discussion below). It is said to be \textbf{numerable} if there is a collection of functions $\phi_i:X \to [0,1]$ such that

\begin{itemize}%
\item $\overline{supp(\phi_i)} \subset U_i$,
\item at each point $x\in X$, only finitely many of the $\phi_i$ are non-zero,
\item $\sum_i \phi_i(x) \equiv 1 \forall x\in X$.

\end{itemize}
The open cover $\phi_i^{-1}(0,1]$ is then a [[locally finite cover]] that [[refinement|refines]] $\{U_i\}$. The functions $\{\phi_i\}$ are a [[partition of unity]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{as_a_coverage_as_a_site}{}\subsubsection*{{As a coverage, as a site}}\label{as_a_coverage_as_a_site}

Numerable open covers form a [[site]] called the \textbf{numerable site}. More precisely, numerable open covers are a [[coverage]] on the category [[Top]] of topological spaces (this is essentially given in \hyperlink{DoldLectures}{Dold's lectures, A.2.17} , but not using this terminology).

For paracompact spaces, numerable covers are cofinal in open covers, so that the numerable site is equivalent to the open cover site for such spaces.

There is also some result by [[Bourbaki]] that [[David Roberts|I]] have to look up that numerable covers are cofinal in [[locally finite open cover|locally finite covers]] of [[normal spaces]].

\hypertarget{relation_to_numerable_bundles}{}\subsubsection*{{Relation to numerable bundles}}\label{relation_to_numerable_bundles}

Many classical theorems concerning [[fiber bundles]] are stated for the numerable site. For example, the [[classifying space]] $\mathcal{B}G$ actually classifies [[bundles]] which trivialise over a numerable cover. (References? Dold for [[Milnor's classifying space]], and tom Dieck I think for Segal's) These are called [[numerable bundles]]. This is because the standard constructions of the universal bundle by Minor and Segal both are trivialisable over a numerable cover.

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

\begin{itemize}%
\item [[Albrecht Dold]], \emph{Partitions of unity in the theory of fibrations}, Ann. of Math. (2) 78 1963 223--255 \href{http://www.ams.org/mathscinet-getitem?mr=155330}{MR0155330} \href{http://www.jstor.org/stable/1970341}{jstor} \href{http://dx.doi.org/10.2307/1970341}{doi}

\end{itemize}
The appendix of

\begin{itemize}%
\item [[Albrecht Dold]], \emph{Lectures on algebraic topology}

\end{itemize}
talks about ``stacked covers'': these are useful for `decomposing' numerable covers of products to a sort of parameterised version depending on a numerable cover of the first factor. This is important in looking at concordance of numerable bundles.

[[!redirects numerable open covers]]

[[!redirects numerable cover]] [[!redirects numerable covers]]



\end{document}