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Homework 3, Real Analysis

Due date: August 25

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

Problem 3

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

Problem 4

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

Problem 5

  1. Let $latex x_k$ be a bounded sequence in the Euclidean space $latex \R^n$. Then it has a convergent subsequence (assume the result in $latex n=1$).

  2. A closed rectangle in $latex \R^n$ is compact.

  3. A closed ball in $latex \R^n$ is compact.