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## Due September 21

### 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 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 $x_n\in X$ is a sequence that converges to $x_0\in Y$, then $x_0\in X$.

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

### Problem 4

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

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