Abstract
The classical Eilenberg-Borsuk theorem on extension of partial mappings into a sphere is generalized to the case of an arbitrary complex 
. It is formulated in terms of extraordinary dimension theory, which is developed in the present paper. When 
is an Eilenberg-MacLane complex, the result can be expressed in terms of cohomological dimension theory. For partial mappings 
of an 
-manifold 
, the following is obtained: Theorem. If 
, then there exists a compactum 
of dimension 
, such that the mapping 
extends to 
and for every abelian group 
with 
the cohomological dimension of 
with coefficients in does not exceed 
. Thus, in comparison with the classical Eilenberg-Borsuk theorem, there is obtained an additional condition as to the cohomological dimension of . Bibliography: 17 titles.