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Since $X$ is second countable and Hausdorff,
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Since $X$ is second countable and Hausdorff,
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we get the upper bound.%
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we get the upper bound.%
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}{Lower bound: $2^{\N} \hookrightarrow X$,
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}{Lower bound: $2^{\N} \hookrightarrow X$,
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upper bound: \nth{2} countable and Hausdorff.
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upper bound: \nth{2} countable and Hausdorff.}
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\end{proof}
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\end{proof}
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\begin{theorem}
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\begin{theorem}
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