Adjunction space

In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another.

be topological spaces, and let

be a continuous map (called the attaching map).

One forms the adjunction space

) by taking the disjoint union of

Formally, where the equivalence relation

consists of the disjoint union of

The topology, however, is specified by the quotient construction.

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.

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.