HW 7

On Tuesday, we discussed continuity of maps, the three equivalent definitions of continuity. Then on Thursday after we reviewed some topologies for metric space, we showed that continuous maps sends a compact set to compact set.

$\gdef\In\subset$

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1. Let $f: X \to Y$ be a map between metric spaces. Check if the following statements are true or not. If you think the statement is false, give some counter example. If you think the statement is true, give some reasoning.

1. If $f$ is a continuous map, then for any open set $U \In X$, the image $f(U)$ is open in $Y$.
2. If $f$ is a continuous map, then for any closed set $E \In Y$, the preimage $f^{-1}(E)$ is closed in $X$.
3. If for any open set $U \In X$, the image $f(U)$ is open in $Y$, then $f$ is continuous.

2. If $X \In Y$ is a subset with the induced metric, and $f: X \to Y$ is the inclusion map, prove that $f$ is continuous. You may use any of the three criterions for checking continuity of $f$.

3. Let $f: (0, \infty) \to \R$ be a map given by $f(x) = \sin(1/x)$, prove that $f$ is continuous. You may use that $\sin(x)$ is a continuous function.

Rudin Ch 4 p99: 1, 2, 4, 6, 7, 18 Note that for problem 18, you only need to prove that $f$ is continuous at every irrational point (since we haven't discuss what is a simple discontinuity.)