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adjunction space
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(Definition)
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Let $X$ and $Y$ be topological spaces, and let $A$ be a subspace of $Y$ . Given a continuous function $\funcdef{f}{A}{X},$ define the space $Z := X\cup_f Y$ to be the quotient space $X\incoprod Y/\sim,$ where the symbol $\incoprod$ stands for disjoint union and the equivalence relation $\sim$ is generated by$$ y \sim f(y)\quad\text{for all}\quad y\in A.$$ $Z$ 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 $Y$ .
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"adjunction space" is owned by antonio.
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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 is 1 reference 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 5540 times total.
Classification:
| AMS MSC: | 54B17 (General topology :: Basic constructions :: Adjunction spaces and similar constructions) |
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Pending Errata and Addenda
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