Lesson plan

Adjunction space

In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be topological spaces, and let A be a subspace of Y . Let f : A \rightarrow X be a continuous map (called the attaching map). One forms the adjunction space X \cup_f Y (sometimes also written as X +_f Y ) by taking the disjoint union of X and Y and identifying a with f(a) for all a in A . Formally, X\cup_f Y = (X\sqcup Y) / \sim where the equivalence relation \sim is generated by a\sim f(a) for all a in A , and the quotient is given the quotient topology. As a set, X \cup_f Y consists of the disjoint union of X and ( Y-A ). The topology, however, is specified by the quotient construction. Intuitively, one may think of Y as being glued onto X via the map f .

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Article summary

In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be topological spaces, and let A be a subspace of Y . Let f : A \rightarrow X be a continuous map (called the attaching map). One forms the adjunction space X \cup_f Y (sometimes also written as X +_f Y ) by taking the disjoint union of X and Y and identifying a with f(a) for all a in A . Formally, X\cup_f Y = (X\sqcup Y) / \sim where the equivalence relation \sim is generated by a\sim f(a) for all a in A , and the quotient is given the quotient topology. As a set, X \cup_f Y consists of the disjoint union of X and ( Y-A ). The topology, however, is specified by the quotient construction. Intuitively, one may think of Y as being glued onto X via the map f .

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  1. Summarise the article in your own words. Which single claim would you verify first, and why?
  2. Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
  3. If two sources disagreed about one of these facts, what would you need in order to trust one over the other?

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