In mathematics, particularly in algebraic topology, a taut pair is a topological pair whose direct limit of cohomology module of open neighborhood of that pair which is directed downward by inclusion is isomorphic to the cohomology module of original pair.
For a topological pair
in a topological space
of such a pair is defined to be a pair such that
respectively.
If we collect all neighborhoods of
, then we can form a directed set which is directed downward by inclusion.
Hence its cohomology module
is a direct system where
is a module over a ring with unity.
If we denote its direct limit by the restriction maps
define a natural homomorphism
is said to be tautly embedded in
(or a taut pair in
is an isomorphism for all
[1] Since the Čech cohomology and the Alexander-Spanier cohomology are naturally isomorphic on the category of all topological pairs,[4] all of the above properties are valid for Čech cohomology.
However, it's not true for singular cohomology (see Example) Let
be the subspace of
which is the union of four sets The first singular cohomology of
and using the Alexander duality theorem on
lim →
varies over neighborhoods of
lim →
is not a monomorphism so that
is not a taut subspace of
with respect to singular cohomology.
is closed in
, it's taut subspace with respect to Alexander cohomology.