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@@ -8,7 +8,7 @@
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\item For any $x \in X$, there exists $K \in \cn(x)$ compact.
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\item For any $x \in X$, $\cn(x)$ admits a fundamental system of neighbourhoods consisting of compact sets.
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\item For any $x \in X$, $\cn(x)$ admits a fundamental system of neighbourhoods consisting of precompact sets.
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\end\{enumerate\}
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\end{enumerate}
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If the above holds, then $X$ is a \textbf{locally compact Hausdorff (LCH)} space.
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\end{definition}
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@@ -122,7 +122,7 @@
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\item[(a)] For each $0 \le k \le n$, $U_k$ is a precompact open set.
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\item[(b)] For each $0 \le k < n$, $\overline{U_k} \subset U_{k+1}$.
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\item[(c)] For each $1 \le k \le n$, $U_k \supset \bigcup_{j = 1}^k K_j$.
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\end\{enumerate\}
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\end{enumerate}
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By \autoref{lemma:lch-compact-neighbour}, there exists $U_{n+1} \in \cn^o(\overline{U_n} \cup K_{n+1})$ precompact. In which case, by (c),
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\[
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@@ -185,7 +185,7 @@
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\begin{enumerate}
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\item[(a)] $\ol{E} \subset \bigcup_{x \in X_E}N_x$.
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\item[(b)] For every $x \in X_E$, $N_x \cap E \ne \emptyset$.
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\end\{enumerate\}
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\end{enumerate}
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Let $X_{\ce} = \bigcup_{E \in \ce}X_\ce$, and for each $E \in \ce$, let
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\[
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