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\begin{document}

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\section*{paracompact topological space}

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

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

[[!include topology - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{colimits}{Colimits}\dotfill \pageref*{colimits} \linebreak
\noindent\hyperlink{HomotopyAndCohomology}{Homotopy and Cohomology}\dotfill \pageref*{HomotopyAndCohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Recall:

\begin{defn}
\label{LocallyFiniteCover}\hypertarget{LocallyFiniteCover}{}
\textbf{([[locally finite cover]])}

Let $(X,\tau)$ be a [[topological space]].

An [[open cover]] $\{U_i \subset X\}_{i \in I}$ of $X$ is called \emph{locally finite} if for each point $x \in X$, there exists a [[neighbourhood]] $U_x \supset \{x\}$ such that it [[intersection|intersects]] only finitely many elements of the cover, hence such that $U_x \cap U_i \neq \emptyset$ for only a [[finite number]] of $i \in I$.

\end{defn}
\begin{defn}
\label{RefinementOfOpenCovers}\hypertarget{RefinementOfOpenCovers}{}
\textbf{([[refinement]] of [[open covers]])}

Let $(X,\tau)$ be a [[topological space]], and let $\{U_i \subset X\}_{i \in I}$ be a [[open cover]].

Then a \emph{[[refinement]]} of this open cover is a set of open subsets $\{V_j \subset X\}_{j \in J}$ which is still an [[open cover]] in itself and such that for each $j \in J$ there exists an $i \in I$ with $V_j \subset U_i$.

\end{defn}
Now:

\begin{defn}
\label{ParacompactSpace}\hypertarget{ParacompactSpace}{}
\textbf{(paracompact topological space)}

A [[topological space]] $(X,\tau)$ is called \textbf{paracompact} if every [[open cover]] of $X$ has a [[refinement]] (def. \ref{RefinementOfOpenCovers}) by a [[locally finite open cover]] (def. \ref{LocallyFiniteCover}).

\end{defn}
\begin{remark}
\label{DifferingTerminology}\hypertarget{DifferingTerminology}{}
\textbf{(differing terminology)}

As with the concept of [[compact topological spaces]] (\href{compact#space#DifferingTerminology}{this remark}), some authors demand a paracompact space to also be a [[Hausdorff topological space]]. See at \emph{[[paracompact Hausdorff space]]}.

\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
Every [[compact space]] is paracompact.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
Every [[locally connected topological space|locally connected]] [[locally compact topological group|locally compact]] [[topological group]] is paracompact (\href{topological+group#ConnectedLocallyCompactTopologicalGroupsAreSigmaCompact}{this prop.}).

\end{example}
\begin{prop}
\label{}\hypertarget{}{}
[[locally compact space|locally compact]] and [[second-countable space|second-countable]] [[Hausdorff space]] are paracompact.

\end{prop}
\begin{prop}
\label{ParacompactFromLocallyCompactAndSigmacompact}\hypertarget{ParacompactFromLocallyCompactAndSigmacompact}{}
[[locally compact and sigma-compact spaces are paracompact]]

\end{prop}
\begin{example}
\label{ParacompactEuclideanSpace}\hypertarget{ParacompactEuclideanSpace}{}
\textbf{([[Euclidean space]] is paracompact)}

For $n \in \mathbb{N}$, then the [[Euclidean space]] $\mathbb{R}^n$, regarded with its [[metric topology]] is paracompact.

\end{example}
\begin{proof}
Euclidean space is evidently [[locally compact topological space|locally compact]] and [[sigma-compact topological space|sigma-compact]]. Therefore the statement follows since [[locally compact and sigma-compact spaces are paracompact]] (prop. \ref{ParacompactFromLocallyCompactAndSigmacompact}).

\end{proof}
\begin{prop}
\label{ParacompactnessPreservedByDisjointUnion}\hypertarget{ParacompactnessPreservedByDisjointUnion}{}
Paracompactness is preserved by forming [[disjoint union spaces]] ([[coproducts]] in [[Top]]).

\end{prop}
\begin{proof}
Consider a disjoint union $X = \coprod X_\lambda$ whose components are paracompact. As the union is disjoint, the components, that is to say, the $X_\lambda$, are open in $X$. Thus any open cover, say $\mathcal{U}$, of $X$ has a refinement by open sets, say $\mathcal{V}$, such that each $V \in \mathcal{V}$ is contained in some $X_\lambda$. Thus we can write $\mathcal{V} = \coprod \mathcal{V}_\lambda$. As each $X_\lambda$ is paracompact, each $\mathcal{V}_\lambda$ has a locally finite refinement, say $\mathcal{W}_\lambda$. Then let $\mathcal{W} := \coprod \mathcal{W}_\lambda$. As each $\mathcal{W}_\lambda$ is a refinement of the corresponding $\mathcal{V}_\lambda$, $\mathcal{W}$ is a refinement of $\mathcal{V}$, and hence of $\mathcal{U}$. As each point of $X$ has a neighbourhood which meets only elements of \emph{one} of the $\mathcal{W}_\lambda$, and as that $\mathcal{W}_\lambda$ is locally finite, $\mathcal{W}$ is locally finite. Thus $\mathcal{U}$ has a locally finite refinement.

\end{proof}
\begin{itemize}%
\item [[manifolds]]

\begin{itemize}%
\item finite-dimensional manifolds are locally compact, so we have the results above, but we also have some converses:

\begin{itemize}%
\item a finite-dimensional Hausdorff topological [[manifold]] is paracompact precisely if it is [[metric space|metrizable]]


\item a finite-dimensional Hausdorff topological manifold is paracompact precisely if each component is second-countable



\end{itemize}

\item [[infinite-dimensional manifolds]] are generally not locally compact, but we still have some results:

\begin{itemize}%
\item The [[Frechet manifold|Frechet]] [[smooth loop space]] of a compact finite dimensional manifold is paracompact.


\item More generally, if $E$ is the sequential [[limit]] of separable Hilbert spaces $H_n$, such that the canonical projections

\begin{displaymath}
p_n : E \to H_n
\end{displaymath}
satisfy

\begin{displaymath}
closure(p_n^{-1}(B)) = p_n^{-1}(closure(B))
\end{displaymath}
for any open ball $B$ in $H_n$, then $E$ is paracompact, and furthermore admits \emph{smooth} partitions of unity.

( \hyperlink{Brylinski}{Brylinski, section I.4})

[[Urs Schreiber]]: don't we need some extra assumption here? Otherwise why wouldn't this imply that every space modeled on $\mathbb{R}^n$ is paracompact, while it is only the second-countable such that are?

\emph{Toby}: Probably Brylinski has a requirement of metrisability or something.

[[Andrew Stacey]]: More generally, the coproduct of paracompact spaces is again paracompact.

Spivak, in \emph{A Comprehensive Introduction to Differential Geometry, I} has an appendix on non-metrisable manifolds. He starts (in the appendix) by defining a manifold to be a Hausdorff locally Euclidean space. Then he proves that for any manifold $M$ TFAE:

\begin{enumerate}%
\item Each component of $M$ is $\sigma$-compact.
\item Each component of $M$ is second countable.
\item $M$ is metrisable.
\item $M$ is paracompact.

\end{enumerate}
The bit to highlight is the words ``Each component of \ldots{}'' in the first two.

\emph{Toby}: Right, and I've got the `each component' clause in the section on finite-dimensional manifolds. But how does this work for infinite-dimensional manifolds? (I'm also unsure how things work for non-Hausdorff manifolds, or more generally for non-Hausdorff locally compact spaces.

[[David Roberts]]: Lang has the result that second countable manifolds are paracompact. For him manifold seems to mean that the space is Hausdorff and locally euclidean, with no restriction on the type of vector space. Further, smooth partitions of unity exist. This is Corollary 3.4 in the 2002 edition of \emph{Introduction to differentiable manifolds}

[[Andrew Stacey]] Surely ``locally Euclidean'' implies that the model space is $\mathbb{R}^n$. Is this the book where he deals with finite and infinite dimensional manifolds all in one go? In that case, I would caution that if I remember right (don't have the book in front of me) he's using manifolds modelled on at most Banach spaces. For smooth partitions of unity to exist then you need to know that there is a smooth bump function, which hinges on some properties of the norm. Kriegl and Michor (who else?) address this in chapter III of their book. In particular, they quote a result due to Kurzweil (1954) that $C([0,1])$ and $\ell^1$ are not $C^1$-regular.

To Toby, I guess that the issue about components for infinite dimensional manifolds is dealt with in what I did before. The question reduces to figuring out if a specific component is paracompact. To use ``metrisable implies paracompact'' you've got to be on a Frechet manifold (since Frechet is the limit of metrisability). Brylinski's construction outlined about is sufficient to ensure that (countable family of semi-norms). To have smooth partitions of unity, you then need smoothly regular. If the semi-norms are smooth (away from zero) then, obviously, that's sufficient. In Brylinski's construction then that comes from the fact that the semi-norms are defined by Hilbertian norms. I'm not sure what the condition on the closures is for, can anyone scan through the proof and see why they are used? For the proof of paracompactness and partitions of unity then I don't see immediately why they are needed. I expect I'm being [[dense]] but if someone could quickly enlighten me, I'd be grateful.

(Incidentally, since this query box is contained within a list, it's important that all paragraphs are indented properly, otherwise strange things happen)

[[David Roberts]]: When I said locally Euclidean, I was being lazy. I meant modelled on some vector space. And yes Lang does finite and infinite dimensions at the same time. Regarding the condition Brylinski uses, he says it implies that the image of $p_n : E \to H_n$ from the ILH space $E$ to each of the Hilbert spaces $H_n$ in the sequence is dense. I don't know how this is used, I'm relying on G00gle books. Ah, but now that I check, Lang also says \emph{unless specified, vector space will mean finite dimensional vector space} \ldots{}

[[Andrew Stacey]] Hmm, I guess I'll have to get Lang and Brylinski out of the library to see exactly what's going on here. I don't see how density can make any difference since we could always restrict to the closure of $p_n(E)$ in $H_n$.



\end{itemize}


\end{itemize}

\item [[CW-complexes are paracompact Hausdorff spaces]] (\hyperlink{Miyazaki52}{Miyazaki 52}), see for instance \hyperlink{Hatcher}{Hatcher, appendix of section 1.2}. For a discussion that highlights which [[axiom of choice|choice principles]] are involved, see (\hyperlink{FP}{Fritsch-Piccinini 90, Theorem 1.3.5 (p. 29 and following)}).


\item metric spaces

\begin{itemize}%
\item every [[separable space|separable]] [[metric space]] is paracompact;


\item every [[metric space]] whatsoever is paracompact, assuming the [[axiom of choice]]; see at \emph{[[metric spaces are paracompact]]}


\item pseudometric spaces are paracompact under the same conditions, if one does not require Hausdorffness;



\end{itemize}

\item In particular we have the following implications

\begin{itemize}%
\item [[second-countable space]] and regular Hausdorff space $\Rightarrow$ [[metric space|metrizable space]] $\Rightarrow$ paracompact space

(the first is Urysohn's metrization theorem, the second is due to \hyperlink{Stone48}{Stone 48}, see also at \emph{[[second-countable regular spaces are paracompact]]} and \emph{[[metric spaces are paracompact]]})


\item paracompact space and locally metric space $\Rightarrow$ [[metric space|metrizable space]]

(this is due to Smirnov)



\end{itemize}

\item special cases

\begin{itemize}%
\item the [[Sorgenfrey line]] is a good example of a paracompact space that doesn't fit into other general classes of paracompact spaces (in particular, it is not [[metrisable topological space|metrisable]], locally compact, or a manifold);

\end{itemize}

\item counterexamples

\begin{itemize}%
\item the [[long line]] is \emph{not} paracompact, even though it is a [[manifold]] (unless one specifically requires paracompactness of manifolds) but it fails to be [[second-countable space|second-countable]] (even though it is connected) or metrisable.


\item the [[Sorgenfrey plane]] (a product of two [[Sorgenfrey lines]]) is not paracompact. This shows that the product of paracompact spaces need not be paracompact.



\end{itemize}


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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item \textbf{Dieudonne's theorem}: [[paracompact Hausdorff spaces are normal]]


\item every paracompact finite-dimensional [[manifold]] has a [[partition of unity]]


\item [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]]

Care should be taken as to which category one constructs partitions of unity on paracompact spaces. For example, analytic partitions of unity generally do not exist on smooth (finite dimensional) manifolds, even when smooth ones do.


\item For paracompact spaces, [[numerable open cover|numerable open covers]] are cofinal in all open covers (in $Top$).


\item [[Michael's theorems]]



\end{itemize}
\begin{example}
\label{CountableCoverOfUnionsofOpenSubsetsInsideGivenCover}\hypertarget{CountableCoverOfUnionsofOpenSubsetsInsideGivenCover}{}
Let $X$ be a [[paracompact Hausdorff space]], and let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]]. Then there exists a [[countable cover]]

\begin{displaymath}
\{V_n \subset X\}_{n \in \mathbb{N}}
\end{displaymath}
such that each element $V_n$ is a [[union]] of [[open subsets]] of $X$ each of which is contained in at least one of the elements $U_i$ of the original cover.

\end{example}
(e.g. \hyperlink{Hatcher}{Hatcher, lemma 1.21})

\begin{proof}
Let $\{f_i \colon  X \to [0,1]\}_{i \in I}$ be a [[partition of unity]] subordinate to the original cover, which exists since [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]].

For $J \subset I$ a [[finite set]], let

\begin{displaymath}
V_J
    \;\coloneqq\;
  \left\{
    x \in X 
    \;\vert\;
    \underset{j \in J}{\forall}
    \left(
       \underset{k \in I \setminus J}{\forall}
       \left(
         f_j(x) \gt f_k(x)
       \right)
    \right)
  \right\}
  \,.
\end{displaymath}
By local finiteness there are only a [[finite number]] of $f_k(x)$ greater than zero, hence the condition on the right is a finite number of strict inequalities. Since the $f_i$ are continuous, this implies that $V_J$ is an [[open subset]].

Moreover, $V_J$ is contained in $supp(f_j)$ for $j \in J$ and hence in one of the $U_i$.

Now for $n \in \mathbb{N}$ take

\begin{displaymath}
V_n 
    \;\coloneqq\;
  \underset{ {J \subset I} \atop { {\vert J\vert} = n } }{\cup}
  V_J
\end{displaymath}
to be the union of the $V_J$ over all subset $J$ with precisely $n$ elements.

The set $\{V_n \subset X\}_{n \in \mathbb{N}}$ is a cover because for any $x \in X$ we have $x \in V_{J_x}$ for

\begin{displaymath}
J_x \coloneqq \{ i \in I \;\vert\; f_i(x) \gt 0  \}
\end{displaymath}
(which is finite by local finitness of the partition of unity).

\end{proof}
\hypertarget{colimits}{}\subsubsection*{{Colimits}}\label{colimits}

See at \emph{[[colimits of paracompact Hausdorff spaces]]}.

\hypertarget{HomotopyAndCohomology}{}\subsubsection*{{Homotopy and Cohomology}}\label{HomotopyAndCohomology}

\begin{itemize}%
\item On paracompact spaces, abelian [[Čech cohomology]] does compute [[abelian sheaf cohomology]],

i.e. the canonical morphism $\check{H}(X,A) \to H(X,A)$ for $A$ any [[chain complex]] of [[sheaf|sheaves]] is an [[isomorphism]] when the [[topological space]] underlying $X$ is paracompact.



\end{itemize}
\begin{prop}
\label{}\hypertarget{}{}
On a paracompact space $X$, every [[hypercover]] of finite height is refined by the [[Cech nerve]] of an ordinary [[open cover]].

\end{prop}
For a hypercover of height $n \in \mathbb{N}$, this follows by intersecting the open covers that are produced by the following lemma for $0 \leq k \leq n$.

\begin{lemma}
\label{}\hypertarget{}{}
For $X$ a paracompact topological space, let $\{U_\alpha\}_{\alpha \in A}$ be an [[open cover]], and let each $(k+1)$-fold intersection $U_{\alpha_0, \cdots, \alpha_{k}}$ be equipped itself with an open cover $\{V^{\alpha_0, \cdots, \alpha_k}_{\beta}\}$.

Then there exists a refinement $\{U'_{\alpha'}\}$ of the original cover, such that each $(k+1)$-fold intersection $U'_{\alpha'_0, \cdots, \alpha'_k}$ for all indices distinct is contained in one of the $V_\beta$.

\end{lemma}
This appears as ([[Higher Topos Theory|HTT, lemma 7.2.3.5]]).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[countably paracompact topological space]]


\item [[compact topological space]], [[countably compact topological space]], [[locally compact topological space]], [[strongly compact topological space]], [[sequentially compact topological space]]



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

The notion of paracompact space was introduced in

\begin{itemize}%
\item [[Jean Dieudonné]], \emph{Une g\'e{}n\'e{}ralisation des espaces compacts}, Journal de Math\'e{}matiques Pures et Appliqu\'e{}es, Neuvi\`e{}me S\'e{}rie, 23: 65--76 (1944)

\end{itemize}
That [[fully normal spaces are equivalently paracompact]] is due to

\begin{itemize}%
\item A. H. Stone, \emph{Paracompactness and product spaces}, Bull. Amer. Math. Soc. Volume 54, Number 10 (1948), 977-982. (\href{http://projecteuclid.org/euclid.bams/1183512390}{Euclid})

\end{itemize}
General accounts include

\begin{itemize}%
\item R. Engelking, \emph{General topology}, chapter 5 is dedicated to paracompact spaces


\item [[Brian Conrad]], \emph{Paracompactness and local compactness}, \href{http://math.stanford.edu/~conrad/diffgeomPage/handouts/paracompact.pdf}{pdf}


\item D. K. Burke, \emph{Covering properties}, in: [[K. Kunen]], J.E. Vaughan (eds.), Handbook of Set-Theoretic Topology, North-Holland (1984) Ch. 9, 347--422


\item [[Alan Hatcher]], section 1.2 of \emph{Vector bundles \& K-theory} (\href{https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}{web})


\item English Wikipedia: \href{http://en.wikipedia.org/wiki/Paracompact_space}{paracompact space}


\item Springer eom: \href{http://eom.springer.de/p/p071300.htm}{paracompact space}, \href{http://eom.springer.de/p/p071310.htm}{paracompactness criteria}


\item Some properties of paracompact spaces are listed and proven in \href{http://www.helsinki.fi/~hjkjunni/top9.pdf}{http://www.helsinki.fi/{\tt \symbol{126}}hjkjunni/top9.pdf}



\end{itemize}
A basic discussion with an eye towards [[abelian sheaf cohomology]] and abelian [[Čech cohomology]] is around page 32 of

\begin{itemize}%
\item [[Jean-Luc Brylinski]], \emph{Loop spaces, characteristic classes geoemetric quantization}

\end{itemize}
\begin{itemize}%
\item [[Rudolf Fritsch]], Renzo A. Piccinini, \emph{Cellular structures in topology}, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). (\href{https://epub.ub.uni-muenchen.de/4493/1/4493.pdf}{pdf})

\end{itemize}
Discussion of paracompactness of [[CW-complexes]] includes

\begin{itemize}%
\item Hiroshi Miyazaki, \emph{The paracompactness of CW-complexes}, Tohoku Math. J. (2) Volume 4, Number 3 (1952), 309-313. 1952 \href{https://projecteuclid.org/euclid.tmj/1178245380}{Euclid}

\end{itemize}
[[!redirects paracompact]] [[!redirects paracompactness]] [[!redirects Paracompact space]] [[!redirects Paracompact spaces]] [[!redirects paracompact space]] [[!redirects paracompact spaces]] [[!redirects paracompact topological space]] [[!redirects paracompact topological spaces]] [[!redirects paracompactum]] [[!redirects paracompactums]] [[!redirects paracompacta]]

[[!redirects Dieudonne's theorem]] [[!redirects Dieudonne's theorem]] [[!redirects Dieudonne's theorem]] [[!redirects Dieudonne theorem]] [[!redirects Dieudonné's theorem]] [[!redirects Dieudonné's theorem]] [[!redirects Dieudonné's theorem]] [[!redirects Dieudonné theorem]]



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