Homework 6, Real Analysis

Due date: September 15

Problem 1

State whether the following are true:

  1. \overline{A\cup B} \subset \overline{A}\cup \overline{B};
  2. \overline{A\cup B} \supset \overline{A}\cup \overline{B};
  3. \overline{A\cap B} \subset \overline{A}\cap \overline{B}; and
  4. \overline{A\cap B} \supset \overline{A}\cap \overline{B}.

Problem 2

The closed ball \bar B_r(x_0) = \{ x\in X: d(x,x_0)\le r\} is a closed set in X.

Problem 3

If f:X\to Y is continuous, its graph G=\{(x,f(x)): x\in X\} is closed in X\times Y.

Problem 4

Give an example of two disjoint closed sets in a metric space at zero distance.

Problem 5

If U\subset \R, then it is the disjoint countable union of open intervals.


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