Adjunction topology

Stub grade: A

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.

Definition

Suppose $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ are topological spaces, $A\in\mathcal{P}(Y)$ is a closed subspace of $Y$ and $f:A\rightarrow X$ is a continuous map, then:

• The adjunction space formed by attaching $Y$ to $X$ along $f$, denoted $X\cup_f Y$ is given by:
  • $$X\cup_f Y=\frac{X\coprod Y}{\langle a\sim f(a)\rangle}$$

    where $X\coprod Y$ denotes the disjoint union of $X$ and $Y$ and $\langle a\sim f(a)\rangle$ denotes the equivalence relation generated by the relation that relates $a$ to the image of $a$ under $f$, considered with the quotient topology.

$f$ is called the attaching map