# 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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