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[parent] a complete subspace of a metric space is closed (Result)

Let $X$ be a metric space, and let $Y$ be a complete subspace of $X$ Then $Y$ is closed.

Proof

Let $x \in \overline Y$ be a point in the closure of $Y$ Then by the definition of closure, from each ball $B(x, \frac 1 n)$ centered in $x$ we can select a point $y_n \in Y$ This is clearly a Cauchy sequence in $Y$ and its limit is $x$ hence by the completeness of $Y$ $x \in Y$ and thus $Y = \overline Y$




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Cross-references: limit, Cauchy sequence, ball, closure, point, proof, closed, subspace, complete, metric space

This is version 2 of a complete subspace of a metric space is closed, born on 2006-12-31, modified 2006-12-31.
Object id is 8704, canonical name is ACompleteSubspaceOfAMetricSpaceIsClosed.
Accessed 2376 times total.

Classification:
AMS MSC54E50 (General topology :: Spaces with richer structures :: Complete metric spaces)

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