a closed subset of a complete metric space is complete


Let X be a complete metric space, and let YX be a closed subset of X. Then Y is completePlanetmathPlanetmathPlanetmathPlanetmath.

Proof

Let {yn}Y be a Cauchy sequenceMathworldPlanetmathPlanetmath in Y. Then by the completeness of X, ynx for some xX. Then every neighborhoodMathworldPlanetmathPlanetmath of x contains points in Y, so xY¯=Y.

Title a closed subset of a complete metric space is complete
Canonical name AClosedSubsetOfACompleteMetricSpaceIsComplete
Date of creation 2013-03-22 16:31:26
Last modified on 2013-03-22 16:31:26
Owner ehremo (15714)
Last modified by ehremo (15714)
Numerical id 4
Author ehremo (15714)
Entry type Result
Classification msc 54E50