  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.
