# 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]}:- [math]X\cup_f Y=\frac{X\coprod Y}{\langle a\sim f(a)\rangle}[/math]
^{[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.

- 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

- [math]X\cup_f Y=\frac{X\coprod Y}{\langle a\sim f(a)\rangle}[/math]

[ilmath]f[/ilmath] is called the *attaching map*^{[1]}

## Notes

- ↑ 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.

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Introduction to Topological Manifolds - John M. Lee