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Homework 6, Real Analysis

Due date: September 15

Problem 1

State whether the following are true:

  1. $latex \overline{A\cup B} \subset \overline{A}\cup \overline{B}$;

  2. $latex \overline{A\cup B} \supset \overline{A}\cup \overline{B}$;

  3. $latex \overline{A\cap B} \subset \overline{A}\cap \overline{B}$; and

  4. $latex \overline{A\cap B} \supset \overline{A}\cap \overline{B}$.

Problem 2

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

Problem 3

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

Problem 4

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

Problem 5

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