Transcription of Problem Set 2: Solutions Math 201A Fall 2016 Problem 1 ...
{{id}} {{{paragraph}}}
Example: confidence
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. Thenx FsinceFis closed , soFiscompact. Alternatively, If{G X: I}is an open cover ofF,then{G : I} Fcis an open cover ofX. SinceXis compact,there is a finite subcover ofXwhich also coversF, soFis compact.
The collection Csatis es the axioms for closed sets in a topological space: (1) ;;R 2C. (2) The intersection of closed sets is closed, since either every set is R and the intersection is R, or at least one set is countable and the intersection in countable, since any subset of a countable set is countable. (3) A nite union of closed sets is closed,
Loading..
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: