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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 1655 times total.

Classification:
AMS MSC: 54E50 (General topology :: Spaces with richer structures :: Complete metric spaces)
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