Compact maps and quasi-finite complexes

The simplest condition characterizing quasi-finite CW complexes K is the implication XτhK⇒β(X)τK for all paracompact spaces X. Here are the main results of the paper:Theorem 0.1 – If {Ks}s∈S is a family of pointed quasi-finite complexes, then their wedge ⋁s∈SKs is quasi-finite.Theorem 0.2 – If K1 and K2 are quasi-finite countable CW complexes, then their join K1*K2 is quasi-finite.Theorem 0.3 – For every quasi-finite CW complex K there is a family {Ks}s∈S of countable CW complexes such that ⋁s∈SKs is quasi-finite and is equivalent, over the class of paracompact spaces, to K.Theorem 0.4 – 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 XτF, where F is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept.