In item 1 here, it is stated that the base-space curvature form $\omega$ of a $U(1)$-principal bundle with connection must have integral periods (i.e. $\omega$'s integration/pairing on integral degree-2 homology classes is integer-valued).
I assume $\omega$ is obtained taking the true curvature $\Omega$ on the total space, realizing it as a real 2-form by "tracing" which is trivial here, then "pushing down" using the Chern-Weil homomorphism.
Would anyone be able to explain / suggest a source explaining the steps for showing $\omega$ has integral periods?
Update: Actually I realized this is an automatic consequence of Chern-Weil theory; see the answer below. Also: it is specifically $\frac{1}{2\pi}\omega$ that must have integral periods.
Apologies, I realized the answer is straightforward.
Specifically, due to Chern-Weil theory we know $\frac{1}{2\pi}\omega$ is a closed form whose de Rham class corresponds to that of the first Chern class, which is an integral cohomology class and therefore has integral periods.
(So it is actually $\frac{1}{2\pi}\omega$ which has integral periods.)
Also, the result of course works for any $G$-bundle, if we take the trace of the curvature.