3.12 Paracompact Spaces
Definition 3.12.1 (Locally Finite).label Let $X$ be a topological space and $\cali{U}\suf \cali{P}(X)$, then $\cali{U}$ is locally finite if for each $x\in X$, there exists $V\in\cali{N}(x)$ such that $\curl{U\in\cali{U}:V\cap U\neq\emp}$ is finite. Let $U\suf X$ and define $\cali{U}\define \curl{\curl{x}:x\in U}$ then we say $U$ is locally finite if $\cali{U}$ is locally finite.
Lemma 3.12.2 (Locally Finite Compact).label Let $X$ be a topological space, $\cali{U}\suf 2^{X}$ locally finite and $K\suf X$ compact then $\curl{U\in\cali{U}:U\cap K\neq \emp}$ is finite.
Proof. For each $x\in K$, there exists $N_{x}\in \cali{N}(x)$ such that $\curl{U\in\cali{U}:U\cap N_x\neq\emp}$ is finite. By compactness of $K$, there exists $X_{K}\suf X$ finite such that $K\suf \cups{}{x\in X_K}N_{x}$. In which case
$\square$
Lemma 3.12.3 (Locally Finite Closure).label Let $X$ be a topological space, $\cali{U}\suf 2^{X}$ locally finite then $\curl{\cl{U}:U\in\cali{U}}$ is locally finite.
Proof. For each $x\in X$, there exists $N_{x}\in\cali{N}^{o}(x)$ such that $\curl{U\in\cali{U}:N_x\cap U\neq\emp}$ is finite. Since $N_{x}$ is open, for any $U\in\cali{U}$ we have
Thus $\curl{U\in\cali{U}:N_x\cap U\neq\emp}=\curl{U\in\cali{U}:N_x\cap\cl{U}\neq\emp}$ is finite.$\square$
Definition 3.12.4 (Refinement).label Let $X$ be a topological space and $\cali{U},\cali{V}\suf\cali{P}(X)$ open covers then $\cali{V}$ is a refinement of $\cali{U}$ if for every $V\in\cali{V}$, there exists $U\in\cali{U}$ such that $V\suf U$.
Definition 3.12.5 (Paracompact).label Let $X$ be a topological space, then $X$ is paracompact if every open cover $X$ admits a locally finite refinement.
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