Abstract: 
A countable CW complex $K$ is quasi-finite (as defined by
A.~Karasev) if for every finite subcomplex $M$ of $K$
there is a finite subcomplex $e(M)$ such that any
map $f:A\to M$, where $A$ is closed in a separable metric space $X$
satisfying $X\tau K$, has an extension $g:X\to e(M)$.
Levin's results imply that none of the
Eilenberg--MacLane spaces $K(G,2)$ is quasi-finite if $G\ne 0$. In
this paper we discuss quasi-finiteness of all Eilenberg--MacLane
spaces. More generally, we deal with CW complexes with finitely many nonzero Postnikov invariants.


Here are the main results of the paper:

\font\sc=cmcsc10\medskip{\sc Theorem 0.1.} {\it 
Suppose $K$ is a countable CW complex
 with finitely many nonzero Postnikov invariants. If $\pi_1(K)$ is a locally finite group and $K$ is quasi-finite, then
$K$ is acyclic.}

\medskip{\sc Theorem 0.2.} {\it 
Suppose $K$ is a countable non-contractible
CW complex
 with finitely many nonzero Postnikov invariants. If $\pi_1(K)$ is nilpotent and $K$ is quasi-finite, then
$K$ is extensionally equivalent to $S^1$.}

1. Fakulteta za Matematiko in Fiziko
   Univerza v Ljubljani
   Jadranska ulica 19
   SI-1111 Ljubljana, Slovenija
   
2. University of Tennessee
   Knoxville, TN 37996, U.S.A.
   
3. Fakulteta za Matematiko in Fiziko
   Univerza v Ljubljani
   Jadranska ulica 19
   SI-1111 Ljubljana, Slovenija
   
4. Fakulteta za Matematiko in Fiziko
   Univerza v Ljubljani
   Jadranska ulica 19
   SI-1111 Ljubljana, Slovenija
   
5. University of Tennessee
   Knoxville, TN 37996, U.S.A.