Alexander-Spanier cohomology
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In mathematics, particularly in algebraic topology Alexander-Spanier cohomology is a cohomology theory arising from differential forms with compact support on a manifold. It is similar to and in some sense dual to de Rham cohomology. It is named for J. W. Alexander and Edwin Henry Spanier (1921-1996).
Given a manifold X, let 
be the real vector space of k-forms on X with compact support, and d be the standard exterior derivative.
Then the Alexander-Spanier cohomology groups 
are the homology of the chain complex 
:
;
i.e., is the vector space of closed k-forms modulo that of exact k-forms.
Despite their definition as the homology of an ascending complex, the Alexander-Spanier groups demonstrate covariant behavior; for example, given the inclusion mapping for an open set U of X, extension of forms on U to X (by defining them to be 0 on X-U) is a map 
inducing a map
.
They also demonstrate contravariant behavior with respect to proper maps - that is, maps such that the inverse image of every compact set is compact. Let f: U → X be such a map; then the pullback
induces a map
.
A Mayer-Vietoris sequence holds for Alexander-Spanier cohomology.
