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. Examples A common example of an adjunction space is given when Y is a closed n-ball (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 1-1 correspondence with the pairs of continuous maps hX : X → Z and hY : Y → Z 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 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 ϕX, ϕY are the maps obtained by composing the quotient map with the canonical injections