The space in question is the Hausdorff topological space with base β:
β = {U(a, b) : a, b ∈ Z, b > 0}, where U(a, b) = {a + kb : k ∈ Z} .
(I have confirmed that this in fact a base of a Hausdorff topology on Z)
I have now been asked to determine whether it is compact. In fact, I know that it is not compact, but do not know how to justify this.
My definition of compact is that a topological space X is compact if every open cover (Ui) with i in I, has a finite subcover. Hence I would appreciate any help in finding an open cover of the space, that has no finite subcover.
HINT: $\mathscr{U}=\{U(0,p):p\text{ is prime}\}$ covers $\Bbb Z\setminus\{-1,1\}$. (Why?) Find open sets $V$ and $W$ containing $-1$ and $1$, respectively, such that $\mathscr{U}\cup\{V,W\}$ has no finite subcover. You may find Dirichlet's theorem useful.