In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let and be topological spaces, and let be a subspace of . Let be a continuous map (called the attaching map). One forms the adjunction space (sometimes also written as ) by taking the disjoint union of and and identifying with for all in . Formally,
where the equivalence relation is generated by for all in , and the quotient is given the quotient topology. As a set, consists of the disjoint union of and (). The topology, however, is specified by the quotient construction.
Intuitively, one may think of as being glued onto via the map .
The continuous maps h : X âª<sub>f</sub> Y → Z are in 1-1 correspondence with the pairs of continuous maps h<sub>X</sub> : X → Z and h<sub>Y</sub> : Y → Z that satisfy h<sub>X</sub>(f(a))=h<sub>Y</sub>(a) for all a in A.
In the case where A is a closed subspace of Y one can show that the map X â X âª<sub>f</sub> Y is a closed embedding and (Y â A) â X âª<sub>f</sub> Y is an open embedding.
The attaching construction is an example of a pushout in the category of topological spaces. That is to say, the adjunction space is universal with respect to the following commutative diagram:
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Here i is the inclusion map and æ<sub>X</sub>, æ<sub>Y</sub> are the maps obtained by composing the quotient map with the canonical injections into the disjoint union of X and Y. One can form a more general pushout by replacing i with an arbitrary continuous map g—the construction is similar. Conversely, if f is also an inclusion the attaching construction is to simply glue X and Y together along their common subspace.