HW 7

This week we proved the equivalence of the two notions of compactness. Here are some more problems

1. If $X$ and $Y$ are open cover compact, can you prove that $X \times Y$ is open cover compact? (try to do it directly, without using the equivalence between open cover compact and sequential compact)

2. Let $f: X \to Y$ be a continuous map between metric spaces. Let $A \In X$ be a subset. Decide if the followings are true or not. If true, give an argument, if false, give a counter-example.

• if $A$ is open, then $f(A)$ is open
• if $A$ is closed, then $f(A)$ is closed.
• if $A$ is bounded, then $f(A)$ is bounded.
• if $A$ is compact, then $f(A)$ is compact.
• if $A$ is connected, then $f(A)$ is connected.

3. Prove that, there is not continuous map $f: [0,1] \to \R$, such that $f$ is surjective. (there is a surjective map from $(0,1) \to \R$ though)