The direct limit
of the cohomology groups with coefficients in an Abelian group
of the nerves of all open coverings
of a topological space
.
The cohomology group of a closed subset
can be defined analogously, using the subsystem
of all such sets from
that have non-empty intersection with
.
The limit of the groups of the pair
defines the cohomology group
of the pair
.
The cohomology sequence
of the pair
is exact, being the limit of the exact cohomology sequences of the pairs of nerves of
.
Aleksandrov–Čech cohomology serves as a substitute for singular cohomology
in general categories of topological spaces, and agrees with it
whenever the applicability of the latter is not in doubt
(specifically, in the case of homologically, locally connected spaces,
in particular locally contractible spaces). It satisfies all
Steenrod–Eilenberg axioms,
and on the category of paracompact spaces it
is uniquely determined by those axioms together with the following conditions: a)
for
;
b) the cohomology group of a discrete union
is naturally isomorphic to the direct product of the cohomology groups of the
;
and c)
for the system of all neighbourhoods
of an arbitrary point
.
Aleksandrov–Čech cohomology is isomorphic to
Alexander–Spanier cohomology.
The latter can be defined with coefficients in a
sheaf, and for paracompact spaces it is isomorphic
to the cohomology defined in sheaf theory.
The approximability of spaces by polyhedra — by the nerves
of closed coverings — was established by
P.S. Aleksandrov
(cf.
[1]–[3]).
For a particular case he has given the definition of an
inverse limit of topological spaces, and, on the basis of
the approximation, the definition of the Betti numbers of metrizable compacta.
The homology groups of compacta are defined in terms of
Vietoris cycles.
L.S. Pontryagin
[4]
introduced direct and inverse spectra of groups, and
applied these notions to the study of the homology groups
of compacta.
E. Čech
began to consider nerves of finite
open coverings of non-compact spaces and on this basis
initiated the homology theory of arbitrary topological spaces. It turned out
later that it is not justifiable to consider only finite
open coverings (since this leads to the rather
complicated homology of the Stone–Čech compactification).
C.H. Dowker
[5]
demonstrated that it is fruitful to use arbitrary open coverings
in the homology and cohomology theory of non-compact spaces.