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hw6
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A one-page homework sheet from David Royster's introductory topology course, MATH 4181 section 001, Fall 1999, kept in the Royster topology notes folder. It has five proof problems: the path components of the topologist's sine curve, path connectedness as a topological invariant, local connectedness and continuous images, locally path connected implying locally connected, and finite subsets being compact.
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MATH 4181 001 Fall 1999
Problem Set 6
1. Identify the two path components of the topologist's sine curve. Show that one of the
components is closed and the other is not.
2. Prove that path connectedness is a topological invariant. Can you prove that it is a
continuous invariant?
3. Prove that local connectedness is a topological invariant, but that the continuous image
of a locally connected set need not be locally connected.
4. Prove that every locally path connected space is locally connected.
5. Prove that every nite subset of a topological space is compact.
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