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A one-page homework sheet from MATH 4181 section 001, Fall 1999, apparently from David Royster's introductory topology course. It has four problems: show a space X is Hausdorff exactly when the diagonal of X×X is closed; show quotient spaces of connected spaces are connected; show quotient spaces of compact spaces are compact; and give a Hausdorff space with a non-Hausdorff quotient.
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MATH 4181 001 Fall 1999
Problem Set 7
1. In the product space XX, the set =f(x;x)jx2Xgis called the diagonal . Prove
that a space Xis Hausdor if and only if the diagonal of XXis a closed set.
2. Prove that if Xis connected, then every quotient space of Xis connected.
3. Prove that if Xis compact, then every quotient space of Xis compact.
4. Give an example of a Hausdor space which has a quotient space that is not Hausdor .
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