A countable CW-complex $K$ is quasi-finite if for any finite subcomplex $M\subset K$, there is (possibly larger) finite subcomplex $e(M)\subset K$, such that for every separable metric space $X$ satisfying
one has a similar property
There is a characterization: a coutable CW-complex $K$ is quasi-finite iff for all separable metric spaces $X$, if $K$ is an absolute extensor of $X$ implies then it is an absolute extensor of its Stone-Čech compactification $\beta(X)$ as well.
In fact, the original definition asks that one has a function $e:M\to e(M)$ (the same under the axiom of choice).
A.Karasev, On two problems in extension theory, arXiv:math.GT/0312269
M.Cencelj, J.Dydak, J.Smrekar, A.Vavpetic, Ž.Virk, Algebraic properties of quasi-finite complexes, Fund. Math. 197 (2007), 67-80 math/0509582