= Solution
If an open set contains an even integer, invariance under every even translation makes it contain all even integers. Similarly, containing an odd integer makes it contain all odd integers. Conversely each parity class is invariant under those translations. Thus the proposed <open sets> are exactly
$$
\boxed{\varnothing,\quad 2\mathbb Z,\quad 2\mathbb Z+1,\quad\mathbb Z.}
$$
They form a <partition topology>. Both the empty and whole sets are included, and any union or finite intersection is again a union of parity classes. Hence \b[this is a <topological space>]. It is not the <discrete topology>: no singleton is open.
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