\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}
