Topological pair

Topological pair

In mathematics, more specifically algebraic topology, a pair (X,A) is short-hand for an inclusion of topological spaces icolon Ahookrightarrow X. Sometimes "i" is assumed to be a cofibration. A morphism from (X,A) to (X',A') is given by two maps fcolon X ightarrow X' and gcolon A ightarrow A' such that gcirc i'=icirc f.

Pairs come up mainly in homology theory and cohomology theory, where chains in "A" are made equivalent to 0, when considered as chains in "X".

Heuristically, one often thinks of a pair "(X,A)" as being akin to the quotient space "X/A".

There is a functor from spaces to pairs, which sends a space "X" to the pair (X,varnothing).

References


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