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 $K_1\ast K_2$ 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 $X\tau {\mathcal F}$, where ${\mathcal F}$ is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept. 
M. Cencelj, J. Dydak, J. Smrekar, A. Vavpetic, and Z. Virk