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@ -42,14 +42,14 @@ Recall the following notions:
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and $U_i \subsetneq X_i$ for only finitely many $i$.
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\end{definition}
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\begin{fact}
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Countable products of separable spaces are separable,
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Countable products of separable spaces are separable.
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\end{fact}
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\begin{definition}
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A topological space $X$ is \vocab{second countable},
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if it has a countable base.
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\end{definition}
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If $X$ is a topological space.
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Then if $X$ is second countable, it is also separable.
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Let $X$ be a topological space.
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If $X$ is second countable, it is also separable.
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However the converse of this does not hold.
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\begin{example}
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@ -81,7 +81,8 @@ However the converse of this does not hold.
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For a metric space, the following are equivalent:
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\begin{itemize}
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\item compact,
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\item \vocab{sequentially compact} (every sequence has a convergent subsequence),
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\item \vocab{sequentially compact}
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(every sequence has a convergent subsequence),
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\item complete and \vocab{totally bounded}
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(for all $\epsilon > 0$ we can cover the space with finitely many $\epsilon$-balls).
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\end{itemize}
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