Topology2007aug - Topology Qualifying Exam August 20, 2007...

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Topology Qualifying Exam August 20, 2007 Do all of the following problems. 1. (20 points) Let X be a set. (a) Define what it means to say that the collection B of subsets of X is a basis for a topology on X . (b) Suppose now that X is endowed with a topology T . Define what it means to say that the subset C of X is compact (c) Let R be the real line. Give a basis for the lower limit topology on R . Let R l denote R with this topology. (d) Show that if C R l is compact, then C is closed and bounded. (e) Show that the converse of (d) is false. 2. (20 points) Let X be a topological space with topology T . (a) Define what it means to say that X is connected . (b) Let D be a subset of X . Define what it means to say D is dense in X . (c) Suppose X is connected and that D X is dense. Let T be the topology on X with subbasis T ∪ { D } . Show that ( X, T ) is connected. 3. (20 points) Recall that a topological space X is said to be completely regular if one-point sets are closed in
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Topology2007aug - Topology Qualifying Exam August 20, 2007...

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