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Consider the subsets A and B of $\mathbb{R}^2$ defined by

$A = \{\left(x,x\sin\frac1x\right):x \in(0,1]\}$ and $B=A \cup \{(0,0)\}.$

Then which of the followings are true?
1. $A$ is compact
2. $A$ is connected
3. $B$ is compact
4. $B$ is connected.

I know that A is connected but not path connected, so 2 is true. But I'm not sure about the others.

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    For 1., note that $(0,0)$ is in the closure of $A$, but not in $A$. For 3., note $B$ is closed and bounded. If you believe that $A$ is connected, you should be able to see that $B$ is also (since $B=\overline A$). (Why is $A$ not path connected?)2012-12-21

3 Answers 3