Tautness (topology)
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. Definition
For a topological pair  in a topological space , a neighborhood  of such a pair is defined to be a pair such that  and  are neighborhoods of  and  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 . The pair  is said to be tautly embedded in  (or a taut pair in ) if  is an isomorphism for all  and . Basic properties
 For pair  of , if two of the three pairs , and  are taut in , so is the third. For pair  of , if  and  have compact triangulation, then  in  is taut. If  varies over the neighborhoods of , there is an isomorphism . If  and  are closed pairs in a normal space , there is an exact relative Mayer-Vietoris sequence for any coefficient module

Properties related to cohomology theory
 Let  be any subspace of a topological space  which is a neighborhood retract of . Then  is a taut subspace of  with respect to Alexander-Spanier cohomology. every retract of an arbitrary topological space is a taut subspace of  with respect to Alexander-Spanier cohomology. A closed subspace of a paracompactt Hausdorff space is a taut subspace of relative to the Alexander cohomology theory

Note
Since the Čech cohomology and the Alexander-Spanier cohomology are naturally isomorphic on the category of all topological pairs, all of the above properties are valid for Čech cohomology. However, it's not true for singular cohomology (see Example)

Dependence of cohomology theory

Example
Let  be the subspace of  which is the union of four sets

The first singular cohomology of  is  and using the Alexander duality theorem on ,  as  varies over neighborhoods of . Therefore,  is not a monomorphism so that  is not a taut subspace of  with respect to singular cohomology. However, since  is closed in , it's taut subspace with respect to Alexander cohomology. See also
Alexander-Spanier cohomology
Čech cohomology

References 

Algebraic topology