Compact maps and quasi-finite complexes

The simplest condition characterizing quasi-finite CW complexes K is 
the implication X\tau_h K implies \beta(X)\tau K for all paracompact spaces X. 
Here are the main results of the paper:

Theorem: If  {K_s}_{s in S} is a family of pointed quasi-finite complexes, 
then their wedge \bigvee\limits_{s in S}K_s$ is quasi-finite.

Theorem:  If K_1 and K_2 are quasi-finite countable complexes, then their join
K_1*K_2 is quasi-finite.

Theorem: For every quasi-finite CW complex K there is a family {K_s}_{s in S} 
of countable CW complexes such that \bigvee\limits_{s in S} K_s is quasi-finite 
and is equivalent, over the class of paracompact spaces, to K.

Theorem: Two quasi-finite CW complexes K and L are equivalent over the class of 
paracompact spaces if and only if they are equivalent over the class of compact 
metric spaces.


Quasi-finite CW complexes lead naturally to the concept of X\tau F, where
F is a family of maps between CW complexes. We generalize some well-known results
of extension theory using that concept.