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

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

Proof

Let $ \{ y_n \} \subseteq Y$ be a Cauchy sequence in $ Y$. Then by the completeness of $ X$, $ y_n \rightarrow x$ for some $ x \in X$. Then every neighborhood of $ x$ contains points in $ Y$, so $ x \in \overline Y = Y$.



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Cross-references: points, contains, neighborhood, Cauchy sequence, proof, closed subset, metric space, complete

This is version 1 of a closed subset of a complete metric space is complete, born on 2006-12-31.
Object id is 8703, canonical name is AClosedSubsetOfACompleteMetricSpaceIsComplete.
Accessed 1695 times total.

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