Lecture 3
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Then let $f(x)$ be the unique point in $X$
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Then let $f(x)$ be the unique point in $X$
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such that
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such that
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\[
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\[
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\{f(x)\} = \bigcap_{n} U_{x \defon n} = \bigcap_{n} \overline{U_{x \defon n}.
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\{f(x)\} = \bigcap_{n} U_{x \defon n} = \bigcap_{n} \overline{U_{x \defon n}}.
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\]
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\]
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(This is nonempty as $X$ is a completely metrizable space.)
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(This is nonempty as $X$ is a completely metrizable space.)
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It is clear that $f$ is injective and continuous.
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It is clear that $f$ is injective and continuous.
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