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Homework 8, Real Analysis

Due October 12

Problem 1

Let X be a metric space. We say that a collection $latex \{F_\alpha\}$ of subsets of X has the finite intersection property (FIP) if any finite subcollection $latex \{F_{\alpha_1}, F_{\alpha_2},\ldots,F_{\alpha_k}\}$ of them has nonempty intersection:

$latex F_{\alpha_1}\cap F_{\alpha_2}\cap\ldots\cap F_{\alpha_k}\not=\emptyset.$

  1. X is compact if and only if the intersection of any collection of closed sets that has the FIP is nonempty.

  2. Give an example of a decreasing sequence of nonempty closed sets in a metric space with empty intersection.

Problem 2

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

  2. Is $latex \bar B_r(x_0) = \overline{B_r(x_0)}$ in every metric space?

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

If X is a metric space, then $latex \diam \bar A = \diam A$ for any $latex A\subset X$.