Alexander-Spanier cohomology with real coefficients is the cohomology of a cochain complex constructed from real-valued
functions on powers of a topological space 
.
For a nonnegative integer 
, let 
be the vector space of
all functions 
, with coboundary
operator
 |
where the hat means that the indicated argument is omitted. Let 
consist of those functions
that vanish on 
for some open
cover 
of 
.
The coboundary operator preserves these subspaces, and Alexander-Spanier cohomology is the cohomology
of the quotient cochain complex 
(Connes and Moscovici 1990, p. 346).
On a smooth manifold 
, these cohomology groups
are naturally isomorphic to de Rham cohomology
(Connes and Moscovici 1990, p. 348), and hence to singular
cohomology with real coefficients (Park, p. 1). Compactness is not required.
These are ordinary cohomology groups, with no
compact support restriction.
