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\textbf{MATH 535, Topology, Homework 5}
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\underline{\textbf{1. (3 (if) +2 (and only if) pts)}}\\
Let $X$ be a topological space. Show that $X$ is a $T_6$ if and only if $X$ is $T_4$ and every closed subset of $X$ is a $G_{\delta}$-set.\\

\textbf{Warning.} Recall that the definition of a $T_6$-space may differ from source to source. Our definition is that $X$ is $T_6$ if $X$ is $T_1$ and for every disjoint closed subsets $A,B \subseteq X$ there exists a continuous function $f: X \rightarrow [0,1]$ with $A=f^{-1}(0)$ and $B=f^{-1}(1)$. Some books use the other part of this iff statement as the definition. The point of this question is to prove the equivalence of these definitions.\\

\underline{\textbf{2. (2 pts)}} Show that every metrizable Lindelöf space is second-countable.\\

\underline{\textbf{3. (3 pts)}} A topological space $X$ is said to be a locally metrizable space if every point has an open neighborhood whose subspace topology is metrizable. Show that if $X$ is a locally metrizable compact Hausdorff space, then $X$ is metrizable.


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\textbf{Solutions}
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