Showing two sets are not homeomorphic I am trying to show two sets are not homeomorphic, but the book I work out of does not give any strategies for this.
My two topologies are Z with standard topology and Z with arithmetic progression topology (with basis B(a, k) = {a + (p^k)m | m ∈ Z} for a fixed integer p).
I think both spaces are Hausdorff (is this true?) so i cannot use that. Any hints?

In $\mathbb Z$ with the standard topology, every subset is open (and closed). That's not the case with $\mathbb Z$ with the arithmetic progression topology. For instance, no finite non-empty set is open.
