= Solution
\b[This does not define a <topology> on $\mathbb Z$.] Every singleton $\{2k\}$ is declared open, but their union over $k\in\mathbb Z$ is $2\mathbb Z$, an infinite proper subset. It is therefore not declared open, violating closure under arbitrary unions. This is the example that <finite subsets do not define a topology on an infinite set>. The empty set, whole space and finite-intersection requirements do hold, but they do not compensate for the failed union axiom. This is different from the <cofinite topology>, which uses finite complements rather than finite sets.
Back to article page
