A classifying space 
for a discrete group 
is a connected CW complex with
fundamental group 
and contractible universal
cover. It is an Eilenberg-Mac Lane space
of type 
:
its higher homotopy groups vanish.
A group has type 
if it admits such a space with finite 
-skeleton, and type 
if it admits one with finitely many cells in every dimension. Type 
allows the classifying space to have infinitely many
dimensions.
