Adjunction space

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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 X and Y be topological spaces, and let A be a subspace of Y. Let f : AX be a continuous map (called the attaching map). One forms the adjunction space XfY (sometimes also written as X +fY) by taking the disjoint union of X and Y and identifying a with f(a) for all a in A. Formally,


where the equivalence relation ~ is generated by a ~ f(a) for all a in A, and the quotient is given the quotient topology. As a set, XfY consists of the disjoint union of X and (YA). The topology, however, is specified by the quotient construction.

Intuitively, one may think of Y as being glued onto X via the map f.



The continuous maps h : XfYZ are in 1-1 correspondence with the pairs of continuous maps hX : XZ and hY : YZ that satisfy hX(f(a))=hY(a) for all a in A.

In the case where A is a closed subspace of Y one can show that the map XXfY is a closed embedding and (YA) → XfY is an open embedding.

Categorical description

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:


Here i is the inclusion map and ϕX, ϕY 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 gthe 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.

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