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adjunction space (Definition)

Let $ X$ and $ Y$ be topological spaces, and let $ A$ be a subspace of $ Y$. Given a continuous function $ f:A\rightarrow X,$ define the space $ Z := X\cup_f Y$ to be the quotient space $ X\amalg Y/\sim,$ where the symbol $ \amalg $ stands for disjoint union and the equivalence relation $ \sim$ is generated by

$\displaystyle y \sim f(y)$   for all$\displaystyle \quad y\in A. $
$ Z$ is called an adjunction of $ Y$ to $ X$ along $ f$ (or along $ A$, if the map $ f$ is understood). This construction has the effect of gluing the subspace $ A$ of $ Y$ to its image in $ X$ under $ f.$
Remark 1   Though the definition makes sense for arbitrary $ A$, it is usually assumed that $ A$ is a closed subspace of $ Y$. This results in better-behaved adjunction spaces (e.g., the quotient of $ X$ by a non-closed set is never Hausdorff).
Remark 2   The adjunction space construction is a special case of the pushout in the category of topological spaces. The two maps being pushed out are $ f$ and the inclusion map of $ A$ into $ Y$.



"adjunction space" is owned by antonio.
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See Also: quotient space

Also defines:  adjunction
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Cross-references: inclusion map, category, pushout, Hausdorff, quotient, closed, image, map, generated by, equivalence relation, disjoint union, quotient space, continuous function, subspace, topological spaces
There are 2 references to this entry.

This is version 7 of adjunction space, born on 2003-02-07, modified 2007-09-23.
Object id is 3992, canonical name is AdjunctionSpace.
Accessed 4741 times total.

Classification:
AMS MSC: 54B17 (General topology :: Basic constructions :: Adjunction spaces and similar constructions)
Pending Errata and Addenda
None.
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Two comments on adjunction space by kompik on 2006-02-09 04:53:26
1. I have seen in some books this definition with the assumption that A is closed subspace. (Walker: Stone-Cech compactification, p.267; Engelking: General Topology)
I wasn't able to find a reference, where this assumption is relaxed. (But I would say, whenever I've used this notion, the condition of closedness wasn't necessary.)
Do you know about a book where adjunction space is used without this assumption?

2. Perhaps it could be mentioned in the article, that the adjunction space is pushout in the category of topological spaces.
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