Homework 8, Real Analysis

Due October 12

Problem 1

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

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 \bar B_r(x_0) = \{ x\in X: d(x,x_0)\le r\} is a closed set in X.
  2. Is \bar B_r(x_0) = \overline{B_r(x_0)} in every metric space?

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

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



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