# Finiteness properties of general topological spaces

It is known that all metric compact ANR have the homotopy type of finite CW complexes. Which spaces are homotopy equivalent or finitely dominated by CW complexes of finite type?

Let $X$ be a space, which we may as well assume connected. If $X$ is an ANR, then Milnor gives that $X$ is homotopy equivalent to a countable CW-complex $X'$. The question of when $X'$ is homotopy equivalent to a complex of finite type is addressed in Theorem A of
Note the necessary condition that $\pi_1(X')$ be finitely presented. I suspect there are more modern references building on Wall's work, but I couldn't find any.