diff --git a/refs.bib b/refs.bib index b45cf36..7473ef1 100644 --- a/refs.bib +++ b/refs.bib @@ -247,4 +247,14 @@ year = {2013}, isbn = {978-1-4614-6955-1}, doi = {10.1007/978-1-4614-6956-8} +} + +@book{Munkres, + author = {Munkres, James R.}, + title = {Topology}, + edition = {2nd}, + publisher = {Prentice Hall}, + address = {Upper Saddle River, NJ}, + year = {2000}, + isbn = {0-13-181629-2} } \ No newline at end of file diff --git a/src/topology/main/lch.tex b/src/topology/main/lch.tex index 50f2667..762c31d 100644 --- a/src/topology/main/lch.tex +++ b/src/topology/main/lch.tex @@ -163,7 +163,7 @@ Let $X$ be a LCH space and $\ce \subset 2^X$ be a locally finite relatively compact open cover of $X$, then there exists locally finite relatively compact 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 \autoref{lemma:locally-finite-compact}. Let + $(\bracs{F_E}_{E \in \ce})$: For each $E \in \ce$, $\bracsn{F \in \ce|F \cap \ol E \ne \emptyset}$ is finite by \autoref{lemma:locally-finite-compact}. Let \[ F_E = \bigcup_{\substack{F \in \ce} \\ F \cap \ol E \ne \emptyset}F \] @@ -171,9 +171,11 @@ then $F_E \in \cn(\ol{E})$ is relatively compact. 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 - \[ - \bracs{E \in \ce|N \cap F_E \ne \emptyset} \subset \bigcup_{\substack{F \in \ce \\ F \cap N \ne \emptyset}}\bracs{E \in \ce|F \cap \ol{E} \ne \emptyset} \subset \bigcup_{\substack{F \in \ce \\ F \cap N \ne \emptyset}}\bracs{E \in \ce|\ol{F} \cap \ol{E} \ne \emptyset} - \] + \begin{align*} + \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} + \end{align*} + By \autoref{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. @@ -207,8 +209,8 @@ by \autoref{proposition:closure-finite-union}. \end{proof} -\begin{proposition} -\label{proposition:lch-paracompact} +\begin{theorem} +\label{theorem:lch-paracompact} Let $X$ be a LCH space, then the following are equivalent: \begin{enumerate} \item $X$ is paracompact. @@ -218,7 +220,7 @@ \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} +\end{theorem} \begin{proof} (1) $\Rightarrow$ (2): For each $x \in X$, there exists a relatively compact 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. @@ -231,9 +233,9 @@ then $\mathcal{U}_F$ is a relatively compact 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 relatively compact 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 + Let $\mathcal{V} = \bigcup_{F \in \cf}\mathcal{V}_F$, then $\mathcal{V}$ is a relatively compact open cover of $X$. For any $x \in X$, there exists $N \in \cn(x)$ such that $\bracs{F \in \cf|N \cap G_F \ne \emptyset}$ is finite. Thus \[ - \bracs{V \in \mathcal{V}| N \cap V} \subset \bigcup_{\substack{F \in \cf \\ N \cap G_F \ne \emptyset}}\mathcal{V}_F + \bracs{V \in \mathcal{V}| N \cap V \ne \emptyset} \subset \bigcup_{\substack{F \in \cf \\ N \cap G_F \ne \emptyset}}\mathcal{V}_F \] is finite, and $\mathcal{V}$ is locally finite. @@ -260,21 +262,23 @@ (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 relatively compact open cover of $\mathcal{U}$. \end{proof} -\begin{proposition} -\label{proposition:paracompact-lch-normal} +There is a more general fact that paracompact Hausdorff spaces are normal \cite[Theorem 6.41.1]{Munkres}. However, the above characterisation can be abused to quickly obtain the same result with local compactness. + +\begin{corollary} +\label{corollary:paracompact-lch-normal} Let $X$ be a paracompact LCH space, then $X$ is normal. -\end{proposition} +\end{corollary} \begin{proof} - Let $A, B \subset X$ be disjoint closed sets. By \autoref{proposition:lch-paracompact}, there exists a partition of unity $\seqi{f}$ subordinate to $\bracs{A^c, B^c}$. Let $I = I_A \sqcup I_B$ such that for each $i \in I_A$, $\supp{f_i} \subset B^c$, and for each $i \in I_B$, $\supp{f_i} \subset A^c$. Take $f = \sum_{i \in I_A}f_i$ and $g = \sum_{i \in I_B}f_i$, then since $f|_B = 0$ and $g|_A = 0$, $\bracs{f \ge 2/3} \in \cn_X(A)$ and $\bracs{f \le 1/3} \in \cn_X(B)$. + Let $A, B \subset X$ be disjoint closed sets. By \autoref{theorem:lch-paracompact}, there exists a partition of unity $\seqi{f}$ subordinate to $\bracs{A^c, B^c}$. Let $I = I_A \sqcup I_B$ such that for each $i \in I_A$, $\supp{f_i} \subset B^c$, and for each $i \in I_B$, $\supp{f_i} \subset A^c$. Take $f = \sum_{i \in I_A}f_i$ and $g = \sum_{i \in I_B}f_i$, then since $f|_B = 0$ and $g|_A = 0$, $\bracs{f \ge 2/3} \in \cn_X(A)$ and $\bracs{f \le 1/3} \in \cn_X(B)$. \end{proof} -\begin{proposition} -\label{proposition:lch-sigma-paracompact} +\begin{corollary} +\label{corollary:lch-sigma-paracompact} Let $X$ be a $\sigma$-compact LCH space, then $X$ is paracompact. -\end{proposition} +\end{corollary} \begin{proof} By \autoref{proposition:lch-sigma-compact}, there exists an exhaustion $\seq{U_n} \subset 2^X$ of $X$ by relatively compact 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 \autoref{proposition:lch-paracompact}, $X$ is paracompact. + 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 \autoref{theorem:lch-paracompact}, $X$ is paracompact. \end{proof}