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hw3
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One-page homework sheet from a topology course (MATH 4181 001, Fall 1999), filed with David Royster's topology notes. It has four problems: separating distinct points by disjoint neighborhoods in a metric space, uniqueness of limits of convergent sequences, finite sets having no limit points and so being closed, and the discrete metric, where every subset is open and closed and has no limit points.
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MATH 4181 001 Fall 1999
Problem Set 3
1. (Exercise 6, page 46) Let aandbbe distinct points in a metric space X. Prove that
there are neighborhoods NaandNbofaandbrespectively such that Na\Nb=;.
2. Letfxngbe a sequence in the metric space Xand assume that fxngconverges to
x2X. Prove that this limit is unique, i.e., if limxn=xand limxn=Lthenx=L.
3. Show that a nite subset of a metric space has no limit points and is therefore a closed
set.
4. Let (X;d) be a metric space with the discrete metric. Prove
(a) Every subset of Xis open.
(b) Every subset of Xis closed.
(c) No subset of Xhas a limit point.
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