I'm reading Ravenel's book Nilpotence and periodicity in stable homotopy theory. In section 1.1, it says the set of homotopy classes of maps of maps between compact manifolds or between algebraic varieties over real or complex numbers is countable. I know that the set of homotopy classes between triangulable spaces is countable. I also know that algebraic varieties over real or complex numbers are triangulable while compact manifolds are not always triangulable. So I want to know how to prove that the set of homotopy classes of maps between compact manifolds is countable.
This may be somewhat heavy-handed but it follows from work of Kirby-Siebenmann that any compact manifold is homotopy equivalent to a finite CW-complex (implying what you want because any finite CW-complex is homotopy equivalent to its regular neighborhood in some Euclidean space, which in turn can be triangulated as a finite simplicital complex).
More precisely, let $M$ be a compact manifold, possibly with boundary. If $M$ has dimension $\ge 6$ and empty boundary, then $M$ is homeomorphic to a CW-complex [p.107 of Kirby-Siebenmann's book "Foundations of Topological manifolds"]. In general, fix $k$ such that $W=M\times D^k$ has dimension $\ge 7$. On the same page 107 it is stated that $W$ is homeomorphic to a mapping cylinder of a map $f:\partial W\to X$, where $X$ is a finite CW-complex. As mentioned above $\partial W$ is also a CW complex. The map $f$ can be homotoped to a cellular map, so the mapping cylinder $W$ is homotopy equivalent to a CW-complex as promised.