Transcription of Problem Set 2: Solutions Math 201A Fall 2016 Problem 1 ...
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Problem Set 2: Solutions Math 201A Fall 2016 Problem 1 ...
Problem Set 2: SolutionsMath 201A:Fall 2016 Problem 1.(a) Prove that a closed subset of a complete metric spaceis complete. (b) Prove that a closed subset of a compact metric space iscompact. (c) Prove that a compact subset of a metric space is closed (a) IfF Xis closed and (xn) is a Cauchy sequence inF, then (xn)is Cauchy inXandxn xfor somex XsinceXis complete. Thenx FsinceFis closed, soFis complete. (b) Suppose thatF XwhereFis closed andXis compact. If (xn)is a sequence inF, then there is a subsequence (xnk) that convergestox XsinceXis compact.
(b) Let AˆQ be any subset of the rational numbers with at least two elements. Choose x;y2Awith x6= y. The irrational numbers are dense in R, so there exists z2RnQ such that x<z<y. Let U= (1 ;z) \A; V = (z;1) \A: Then U, V are open sets in the relative topology on A. Moreover, x2U, y2V so U, V are nonempty, and U\V = ;, U[V = A. It
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