Category Archives: Topology

Fall 2004 #1

Problem Statement: (a) Given a set define what it means for a point to be an interior point of . (b) For let . Show that is open in . (a)Definition: A point is an interior point of if there exists … Continue reading

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September 2000

Problem Statement: Use the open cover definition of compactness to show that any finite union of compact sets is compact. Proof: Let be compact sets in . We wish to show that is a compact set. Let be an open cover of … Continue reading

Posted in Analysis, Compact, Topology | Leave a comment

May 1990 #3

Problem Statement: For , let denote the set of cluster points of . Show there exists no set such that . Proof: Suppose that there is a set such that . Let such that is irrational. Let and consider the … Continue reading

Posted in Analysis, Dense, Topology | Leave a comment