# Adjunction topology

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## Definition

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

• The adjunction space[1] formed by attaching [ilmath]Y[/ilmath] to [ilmath]X[/ilmath] along [ilmath]f[/ilmath][1], denoted [ilmath]X\cup_f Y[/ilmath] is given by[1]:
• $X\cup_f Y=\frac{X\coprod Y}{\langle a\sim f(a)\rangle}$[Note 1]
where [ilmath]X\coprod Y[/ilmath] denotes the disjoint union of [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] and [ilmath]\langle a\sim f(a)\rangle[/ilmath] denotes the equivalence relation generated by the relation that relates [ilmath]a[/ilmath] to the image of [ilmath]a[/ilmath] under [ilmath]f[/ilmath], considered with the quotient topology.

[ilmath]f[/ilmath] is called the attaching map[1]

## Notes

1. Some authors use [ilmath]\frac{X\coprod Y}{a\sim f(a)} [/ilmath] or simply just [ilmath]\frac{X\coprod Y}{\sim} [/ilmath] where the relation is understood. I use [ilmath]\langle\cdot\rangle[/ilmath] in line with common notation for generators here.