Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 1 / 12F / a / Solution

Not a topology. Each singleton $\{n\}$ with $n>0$ is finite and hence belongs to the proposed family, but their union $\{1,2,3,\dots \}$ is infinite and excludes zero, so does not belong. The arbitrary-union axiom fails.