In mathematics, particularly in algebraic topology, a taut pair is a topological pair ( A , B ) {\displaystyle (A,B)} , whose cohomology modules H q ( A , B ; G ) {\displaystyle H^{q}(A,B;G)} are isomorphic to the direct limit of the cohomology modules H q ( U , V ; G ) , {\displaystyle H^{q}(U,V;G),} with ( U , V ) {\displaystyle (U,V)} a pair of open neighborhoods of ( A , B ) {\displaystyle (A,B)} , where the direct limit is induced by inclusion.
Definition For a topological pair ( A , B ) {\displaystyle (A,B)} in a topological space X {\displaystyle X} , a neighborhood ( U , V ) {\displaystyle (U,V)} of such a pair is defined to be a pair such that U {\displaystyle U} and V {\displaystyle V} are neighborhoods of A {\displaystyle A} and B {\displaystyle B} respectively. If we collect all neighborhoods of ( A , B ) {\displaystyle (A,B)} , then we can form a directed set which is directed downward by inclusion. Hence its cohomology module H q ( U , V ; G ) {\displaystyle H^{q}(U,V;G)} is a direct system where G {\displaystyle G} is a module over a ring with unity. If we denote its direct limit by
H ¯ q ( A , B ; G ) = lim → H q ( U , V ; G ) {\displaystyle {\bar {H}}^{q}(A,B;G)=\varinjlim H^{q}(U,V;G)}
the restriction maps H q ( U , V ; G ) → H q ( A , B ; G ) {\displaystyle H^{q}(U,V;G)\to H^{q}(A,B;G)} define a natural homomorphism i : H ¯ q ( A , B ; G ) → H q ( A , B ; G ) {\displaystyle i:{\bar {H}}^{q}(A,B;G)\to H^{q}(A,B;G)} . The pair ( A , B ) {\displaystyle (A,B)} is said to be tautly embedded in X {\displaystyle X} (or a taut pair in X {\displaystyle X} ) if i {\displaystyle i} is an isomorphism for all q {\displaystyle q} and G {\displaystyle G} .
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