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Complete characterisation of paracompactness in LCH spaces.

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Bokuan Li
2026-01-20 14:16:09 -05:00
parent 9d2ca2c3e2
commit a1f2477dcf
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src/topology/main/index.tex
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@@ -14,5 +14,6 @@
\input{./src/topology/main/unity.tex}
\input{./src/topology/main/compact.tex}
\input{./src/topology/main/sigma-compact.tex}
\input{./src/topology/main/para.tex}
\input{./src/topology/main/lch.tex}
\input{./src/topology/main/baire.tex}

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src/topology/main/lch.tex
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@@ -92,7 +92,7 @@
\[
U_{n+1} \supset \ol{U_n} \cup K_{n+1} \supset \bigcup_{j = 1}^n K_j \cup K_{n+1} = \bigcup_{j = 1}^{n+1}K_j
\]
Thus $\bracs{U_j}_0^{n+1}$ satisfies (a), (b), and (c), and $\seq{U_n}$ is an exhaution of $X$ by compact sets.
Thus $\bracs{U_j}_0^{n+1}$ satisfies (a), (b), and (c), and $\seq{U_n}$ is an exhaustion of $X$ by compact sets.
\end{proof}
\begin{proposition}[{{\cite[Proposition 4.41]{Folland}}}]
@@ -119,5 +119,105 @@
Therefore $\seqf{g_j}$ is the desired partition of unity.
\end{proof}
\begin{lemma}
\label{lemma:lch-locally-finite-precompact-refine}
Let $X$ be a LCH space and $\ce \subset 2^X$ be a locally finite precompact open cover of $X$, then there exists locally finite precompact open covers $\bracs{F_E}_{E \in \ce}, \bracs{G_E}_{E \in \ce} \subset 2^X$ such that for each $E \in \ce$, $F_E \subset \ol{F_E} \subset E \subset \ol{E} \subset G_E$.
\end{lemma}
\begin{proof}
$(\bracs{F_E}_{E \in \ce})$: For each $E \in \ce$, $\bracs{F \in \ce|F \cap \ol E \ne \emptyset}$ is finite by \ref{lemma:locally-finite-compact}. Let
\[
F_E = \bigcup_{\substack{F \in \ce} \\ F \cap \ol E \ne \emptyset}F
\]
then $F_E \in \cn(\ol{E})$ is precompact.
LOCALLY FINITE EXHAUSTION
Let $N \subset X$ and $E \in \ce$. If $N \cap F_E \ne \emptyset$, then there exists $F \in \ce$ such that $N \cap F \ne \emptyset$ and $F \cap \ol{E} \ne \emptyset$. Thus
\[
\bracs{E \in \ce|N \cap F_E \ne \emptyset} \subset \bigcup_{\substack{F \in \ce \\ F \cap N \ne \emptyset}}\bracs{E \in \ce|F \cap \ol{E} \ne \emptyset} \subset \bigcup_{\substack{F \in \ce \\ F \cap N \ne \emptyset}}\bracs{E \in \ce|\ol{F} \cap \ol{E} \ne \emptyset}
\]
By \ref{lemma:locally-finite-closure}, $\bracsn{\ol E|E \in \ce}$ is also locally finite. Hence for every $F \in \ce$, $\bracsn{E \in \ce|F \cap \ol{E} \ne \emptyset}$ is finite.
Let $x \in X$, then there exists $N \in \cn(x)$ such that $\bracs{F \in \ce|N \cap F \ne \emptyset}$ is finite. In which case, $\bracs{E \in \ce|N \cap F_E \ne \emptyset}$ is finite as well. Therefore $\bracs{F_E}_{E \in \ce}$ is locally finite.
$(\bracs{G_E}_{E \in \ce})$: For each $x \in X$, there exists $E \in \ce$ and $N_x \in \cn^o(x)$ precompact with $x \in N_x \subset \ol{N_x} \subset E$.
For any $E \in \ce$, $\ol{E}$ is compact, so there exists $X_E \subset X$ finite such that
\begin{enumerate}
\item[(a)] $\ol{E} \subset \bigcup_{x \in X_E}N_x$.
\item[(b)] For every $x \in X_E$, $N_x \cap E \ne \emptyset$.
\end{enumerate}
Let $X_{\ce} = \bigcup_{E \in \ce}X_\ce$, and for each $E \in \ce$, let
\[
G_E = \bigcup_{\substack{x \in X_\ce \\ \ol{N_x} \subset E}}N_x
\]
then $\bracs{G_E}_{E \in \ce}$ is an open cover of $X$. Since $G_E \subset E$ for all $E \in \ce$, $\bracs{G_E}_{E \in \ce}$ is locally finite.
It remains to show that $\ol{G_E} \subset E$. Let $x \in X_F$ such that $N_x \subset E$, then $N_x \cap F \ne \emptyset$. Since $N_x \subset E$, $E \cap F \ne \emptyset$. Thus
\[
\bracsn{x \in X_\ce|\ol{N_x} \subset E} \subset \bigcup_{\substack{F \in \ce \\ E \cap F \ne \emptyset}}X_F \subset \bigcup_{\substack{F \in \ce \\ \ol E \cap F \ne \emptyset}}X_F
\]
is finite by \ref{lemma:locally-finite-compact}, so
\[
\ol{G_E} = \bigcup_{\substack{x \in X_\ce \\ \ol{N_x} \subset E}}\ol N_x \subset E
\]
by \ref{proposition:closure-finite-union}.
\end{proof}
\begin{proposition}
\label{proposition:lch-paracompact}
Let $X$ be a LCH space, then the following are equivalent:
\begin{enumerate}
\item $X$ is paracompact.
\item There exists a locally finite precompact open cover $\cf$ of $X$.
\item For any open cover $\mathcal{U}$ of $X$, there exists a locally finite refinement $\mathcal{V}$ of $\mathcal{U}$ consisting of precompact open sets.
\item For any open cover $\mathcal{U}$ of $X$, there exists locally finite refinements $\seqi{V}, \seqi{W} \subset 2^X$ of $\mathcal{U}$ consisting of precompact open sets such that $\ol{W_i} \subset V_i$ for all $i \in I$.
\item For any open cover $\mathcal{U}$ of $X$, there exists a $C_c(X; [0, 1])$ partition of unity subordinate to it.
\item $X$ admits a $C_c(X; [0, 1])$ partition of unity.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) $\Rightarrow$ (2): For each $x \in X$, there exists a precompact open neighbourhood $U_x \in \cn^o(x)$. Since $\bracs{U_x| x \in X}$ is an open cover of $X$, there exists a locally finite refinement $\mathcal{V}$. For each $V \in \mathcal{V}$, there exists $x \in X$ such that $V \subset U_x$. In which case, $\ol{V} \subset \ol{U_x}$ is compact.
(2) $\Rightarrow$ (3): Let $\cf \subset 2^X$ be a locally finite open cover of $X$ consisting of precompact open sets. By \ref{lemma:lch-locally-finite-precompact-refine}, there exists a locally finite open cover $\bracs{G_F}_{F \in \cf}$ of $X$ consisting of precompact open sets such that $\ol{F} \subset G_F$ for all $F \in \cf$.
For each $F \in \cf$, let
\[
\mathcal{U}_F = \bracs{U \cap G_F|U \in \mathcal{U}}
\]
then $\mathcal{U}_F$ is a precompact open cover of $\ol{F}$. By compactness of $\ol{F}$, there exists $\mathcal{V}_F \subset \mathcal{U}_F$ finite such that $\ol{F} \subset \bigcup_{V \in \mathcal{V}_F}V$.
Let $\mathcal{V} = \bigcup_{F \in \cf}\mathcal{V}_F$, then $\mathcal{V}$ is a precompact open cover of $X$. For any $x \in X$, there exists $N \in \cn(x)$ such that $\bracs{F \in \cf|N \cap G_F}$ is finite. Thus
\[
\bracs{V \in \mathcal{V}| N \cap V} \subset \bigcup_{\substack{F \in \cf \\ N \cap G_F \ne \emptyset}}\mathcal{V}_F
\]
is finite, and $\mathcal{V}$ is locally finite.
(3) $\Rightarrow$ (4): By \ref{lemma:lch-locally-finite-precompact-refine}.
(4) $\Rightarrow$ (5): Let $\seqi{V}, \seqi{W} \subset 2^X$ be locally finite refinements of $\mathcal{U}$ consisting of precompact open sets such that for each $i \in I$, $\ol{W_i} \subset V_i$.
By Urysohn's Lemma (\ref{lemma:lch-urysohn}), there exists $\seqi{f} \in C_c(X; [0, 1])$ such that for each $i \in I$, $f_i|_{\ol{W_i}} = 1$ and $\supp{f_i} \subset V_i$.
Let $F = \sum_{i \in I}f_i$. For each $x \in X$, there exists $N_x \in \cn^o(x)$ such that $\bracs{i \in I|N_x \cap V_i \ne \emptyset}$ is finite. In which case,
\[
F|_{N_x} = \sum_{\substack{i \in I \\ N_x \cap V_i \ne \emptyset}}f_i|_{N_x}
\]
thus $F|_{N_x} \in C(N_x; \real)$. By \ref{lemma:gluing-continuous}, $F \in C(X; \real)$.
Since $\seqi{W}$ is an open cover of $X$, $F(x) > 0$ for all $x \in X$. For each $i \in I$, let $g_i = f_i/F$, then $g_i \in C_c(X; [0, 1])$ with $\supp{g_i} = \supp{f_i} \subset W_i$. For any $x \in X$, there exists $N_x \in \cn^o(x)$ such that $\bracs{i \in I|N_x \cap W_i \ne \emptyset}$ is finite. In which case, $\bracs{i \in I|0 < g_i|_{N_x}}$ is also finite. Thus $\seqi{g}$ is a $C_c$ partition of unity subordinate to $\mathcal{U}$.
(5) $\Rightarrow$ (1): Let $\mathcal{U}$ be an open cover of $X$ and $\seqi{f} \subset C_c(X; [0, 1])$ subordinate to $\mathcal{U}$. For each $i \in I$, let $V_i = \bracs{f_i > 0}$, then $\seqi{V}$ is a locally finite refinement of $\mathcal{U}$.
(5) $\Rightarrow$ (6): Take $\mathcal{U} = \bracs{X}$.
(6) $\Rightarrow$ (2): Let $\seqi{f} \subset C_c(X; [0, 1])$ be a partition of unity. For each $i \in I$, let $V_i = \bracs{f_i > 0}$, then $\seqi{V}$ is a locally finite precompact open cover of $\mathcal{U}$.
\end{proof}
\begin{proposition}
\label{proposition:lch-sigma-paracompact}
Let $X$ be a $\sigma$-compact LCH space, then $X$ is paracompact.
\end{proposition}
\begin{proof}
By \ref{proposition:lch-sigma-compact}, there exists an exhaustion $\seq{U_n} \subset 2^X$ of $X$ by precompact open sets. Denote $U_0 = \emptyset$. For each $n \in \natp$, let $V_n = U_{n+1} \setminus \ol{U_{n-1}}$.
Let $x \in X$, then there exists $n \in \natp$ such that $x \in U_n \setminus U_{n-1}$. In which case, if $n > 1$, then $x \in U_{n} \setminus \ol{U_{n - 2}} = V_{n-1}$. If $n = 1$, then $x \in U_{2} = V_1$. Thus $\seq{V_n}$ is an open cover of $X$. In addition, for any $m, n \in \natp$ with $m \le n$, $V_m \cap V_n \ne \emptyset$ implies that $n - m < 2$, so $\seq{V_n}$ is locally finite. By (2) of \ref{proposition:lch-paracompact}, $X$ is paracompact.
\end{proof}

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src/topology/main/para.tex Normal file
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\section{Paracompact Spaces}
\label{section:paracompact}
\begin{definition}[Locally Finite]
\label{definition:locally-finite}
Let $X$ be a topological space and $\mathcal{U} \subset 2^X$, then $\mathcal{U}$ is \textbf{locally finite} if for every $x \in X$, there exists $V \in \cn(x)$ such that $\bracs{U \in \mathcal{U}| V \cap U \ne \emptyset}$ is finite.
\end{definition}
\begin{lemma}
\label{lemma:locally-finite-compact}
Let $X$ be a topological space, $\mathcal{U} \subset 2^X$ be locally finite, and $K \subset X$ compact, then $\bracs{U \in \mathcal{U}|U \cap K \ne \emptyset}$ is finite.
\end{lemma}
\begin{proof}
For each $x \in K$, there exists $N_x \in \cn(x)$ such that $\bracs{U \in \mathcal{U}|U \cap N_x \ne \emptyset}$ is finite. By compactness of $K$, there exists $X_K \subset X$ finite such that $K \subset \bigcup_{x \in X_K}N_x$. In which case,
\[
\bracs{U \in \mathcal{U}|U \cap K \ne \emptyset} \subset \bigcup_{x \in X_K}\bracs{U \in \mathcal{U}|U \cap N_x \ne \emptyset}
\]
\end{proof}
\begin{lemma}
\label{lemma:locally-finite-closure}
Let $X$ be a topological space, $\mathcal{U} \subset 2^X$ be locally finite, then $\bracsn{\ol{U}|U \in \mathcal{U}}$ is also locally finite.
\end{lemma}
\begin{proof}
For each $x \in X$, there exists $N_x \in \cn^o(x)$ such that $\bracs{U \in \mathcal{U}|N_x \cap U \ne \emptyset}$ is finite. Since $N_x$ is open, for any $U \in \mathcal{U}$, $N_x \cap U = \emptyset$ implies that $N_x^c \supset \ol{U}$. Thus $\bracsn{U \in \mathcal{U}|N_x \cap \ol U \ne \emptyset}$ is finite as well.
\end{proof}
\begin{definition}[Refinement]
\label{definition:refinement}
Let $X$ be a topological space and $\mathcal{U}, \mathcal{V} \subset 2^X$ be open covers, then $\mathcal{V}$ is a \textbf{refinement} of $\mathcal{U}$ if for every $V \in \mathcal{V}$, there exists $U \in \mathcal{U}$ such that $V \subset U$.
\end{definition}
\begin{definition}[Paracompact]
\label{definition:paracompact}
Let $X$ be a topological space, then $X$ is \textbf{paracompact} if every open cover of $X$ admits a locally finite refinement.
\end{definition}