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Which of the following topologies on $\mathbb{Z}$ is(are) locally compact?

The topology generated by $\{A\subset\mathbb{Z}:\mathbb{Z}-A\text{ is finite}\}$.

The order topology generated by division.

The order topology generated by $\leq$.

The topology generated by $\{n : |\sin(n)-a|< r \}$, $a,r\in\mathbb{R}$.