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Problem Sets

Table of Contents

Problem Set 0 (not to be turned in)

If you haven't worked with open sets before or need a refresher, try proving these results. These are all standard results, but please try to prove them without looking them up. Do not turn these in; the first graded problem set will be assigned next week.

When \(X\) is a metric space, we equip it with the metric topology unless otherwise specified.

  1. Let \((X,d)\) be a metric space, let \(x\in X\), let \(r>0\). Show that \[B(x,r)=\{y\in X: d(x,y) < r\}\] is an open set.
  2. Show that the metric topology on \(X\) is in fact a topology.
  3. Give an example of a metric space \(X\) and a sequence of open sets \(U_1,\dots, U_n\) such that \(\bigcap_i U_i\) is not an open set.
  4. Let \((X,d)\) be a metric space. Show that every open subset of \(X\) is a union of open balls. (Note that this need not be a countable union of open balls.)
  5. Let \((X,\mathscr{S})\), \((Y,\mathscr{T})\) be topological spaces and let \(f\from X\to Y\) be a continuous map. Show that if \(x_n\to a\), then \(f(x_n) \to f(a)\).

Created: 2026-09-01 Tue 18:15

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