Adjunction space


In mathematics, an adjunction 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. One forms the adjunction space X ∪f Y by taking the disjoint union of X and Y and identifying a with f for all a in A. Formally,
where the equivalence relation ~ is generated by a ~ f 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. 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

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=hY 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 → 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 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.