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. Specifically, let X and Y be topological spaces with A a subspace of Y. Let f : A → X be a continuous map (called the attaching map). One forms the adjunction space X ∪fY by taking the disjoint union of X and Y and identifying x with f(x) for all x in A. Schematically,
Sometimes, the adjunction is written as . Intuitively, one may think of Y as being glued onto X via the map f.
As a set, X ∪fY consists of the disjoint union of X and (Y − A). The topology, however, is specified by the quotient construction. In the case where A is a closed subspace of Y one can show that the map X → X ∪fY is a closed embedding and (Y − A) → X ∪fY 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 following commutative diagram:
...
Wikipedia