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Homework 5, Real Analysis

Due September 21

Problem 1

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

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

Problem 2

Let X be a discrete metric space.

  1. Any function on X is uniformly continuos.

  2. Is X always compact?

  3. If not, give necessary and sufficient conditions for a discrete space to be compact.

Problem 3

Let X be a subspace of Yi.e. a subset of the metric space Y with the same metric. We say X is closed in Y if X contains all its limits: if $latex x_n\in X$ is a sequence that converges to $latex x_0\in Y$, then $latex x_0\in X$.

If Y is complete, then X is complete if and only if X is closed in Y.

Problem 4

  1. Every bounded set in $latex \R^l$ is totally bounded.

  2. Every closed and bounded set in $latex \R^l$ is compact.