A countable CW-complex is quasi-finite if for any finite subcomplex , there is (possibly larger) finite subcomplex , such that for every separable metric space satisfying
one has a similar property
There is a characterization: a coutable CW-complex is quasi-finite iff for all separable metric spaces , if is an absolute extensor of implies then it is an absolute extensor of its Stone-Čech compactification as well.
In fact, the original definition asks that one has a function (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