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Let:

$$A=\{(x,y)\in\mathbb{R}^2:-1\leq x\leq 1,y=\sin(1/x)\}$$

...with the topology induced by Euclidean distance, and let $B$ be $\mathbb{Z}$ with the topology generated by:

$$\{\{n:n>a\}:a\in\mathbb{Z}\}$$

Which of the following is true?

$A$ is compact but $B$ isn't.

Neither is compact.

$B$ is compact but $A$ isn't.

Both are compact.