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 : A → X be a continuous map (called the attaching map). One forms the adjunction space X ∪_{f} Y (sometimes also written as X +_{f} Y) 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, X ∪_{f} Y consists of the disjoint union of X and (Y − A). The topology, however, is specified by the quotient construction.
Intuitively, one may think of Y as being glued onto X via the map f.
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Category Theory II 6.2: FreeForgetful Adjunction, Monads from Adjunctions

The Formal Theory of Adjunctions, Monads, Algebras, and Descent (Emily Riehl @ MSRI)

7 Morphisms
Transcription
Examples
 A common example of an adjunction space is given when Y is a closed nball (or cell) and A is the boundary of the ball, the (n−1)sphere. Inductively attaching cells along their spherical boundaries to this space results in an example of a CW complex.
 Adjunction spaces are also used to define connected sums of manifolds. Here, one first removes open balls from X and Y before attaching the boundaries of the removed balls along an attaching map.
 If A is a space with one point then the adjunction is the wedge sum of X and Y.
 If X is a space with one point then the adjunction is the quotient Y/A.
Properties
The continuous maps h : X ∪_{f} Y → Z are in 11 correspondence with the pairs of continuous maps h_{X} : X → Z and h_{Y} : Y → Z that satisfy h_{X}(f(a))=h_{Y}(a) for all a in A.
In the case where A is a closed subspace of Y one can show that the map X → X ∪_{f} Y is a closed embedding and (Y − A) → X ∪_{f} Y 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 Φ_{X}, Φ_{Y} 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.
See also
References
 Stephen Willard, General Topology, (1970) AddisonWesley Publishing Company, Reading Massachusetts. (Provides a very brief introduction.)
 "Adjunction space". PlanetMath.
 Ronald Brown, "Topology and Groupoids" pdf available , (2006) available from amazon sites. Discusses the homotopy type of adjunction spaces, and uses adjunction spaces as an introduction to (finite) cell complexes.
 J.H.C. Whitehead "Note on a theorem due to Borsuk" Bull AMS 54 (1948), 11251132 is the earliest outside reference I know of using the term "adjuction space".