Extension dimension and quasi-finite CW-complexes

We extend the definition of quasi-finite complexes from countable complexes to arbitrary ones and provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications. Several results related to the class of quasi-finite complexes are established, such as completion of metrizable spaces, existence of universal spaces and a version of the factorization theorem. Furthermore, we define UV ( L ) -spaces in the realm of metrizable spaces and show that some properties of UV ( n ) -spaces and UV ( n ) -maps remain valid for UV ( L ) -spaces and UV ( L ) -maps, respectively.