Homework 3, Real Analysis

Due date: August 25

Problem 1

Let X be a compact space and f:X\to Y a continuous bijection. Then f^{-1}:Y\to X is continuous.

Give an example of a continuous bijection f:X\to Y, for a noncompact X, whose inverse is not continuous.

Problem 2

Let (X,d_1), (X,d_2) have the same convergent sequences. Then (X,d_1) is compact if and only if (X,d_2) is compact.

Problem 3

Let x_n be a Cauchy sequence and x_{n_k}\to x. Then x_n \to x.

Problem 4

Give necessary and sufficient conditions for the discrete space (X,d) to be compact.

Problem 5

  1. Let x_k be a bounded sequence in the Euclidean space \R^n. Then it has a convergent subsequence (assume the result in n=1).
  2. A closed rectangle in \R^n is compact.
  3. A closed ball in \R^n is compact.
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