Compact maps and quasi-finite complexes

The simplest condition characterizing quasi-finite CW complexes K is the implication X τ h K ⇒ β ( X ) τ K for all paracompact spaces X. Here are the main results of the paper: m 0.1 s } s ∈ S is a family of pointed quasi-finite complexes, then their wedge ⋁ s ∈ S K s is quasi-finite. m 0.2 and K 2 are quasi-finite countable CW complexes, then their join K 1 * K 2 is quasi-finite. m 0.3 ery quasi-finite CW complex K there is a family { K s } s ∈ S of countable CW complexes such that ⋁ s ∈ S K s is quasi-finite and is equivalent, over the class of paracompact spaces, to K. m 0.4 asi-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. finite CW complexes lead naturally to the concept of X τ F , where F is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept.