Lesson plan
Pointed set
In mathematics, a pointed set (also based set or rooted set ) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point (also spelled basepoint). A map between pointed a sets (X, x_0) and (Y, y_0) —called a based map, pointed map, or point-preserving map —is a function from X to Y that maps one basepoint to another, i.e. a map f : X \to Y such that . A based map is usually denoted . Pointed sets are very simple algebraic structures. In the sense of universal algebra, a pointed set is a set X together with a single nullary operation , which picks out the basepoint. Pointed maps are the homomorphisms of these algebraic structures. The class of all pointed sets together with the class of all based maps forms a category. Every pointed set can be converted to an ordinary set by forgetting the basepoint (the forgetful functor is faithful), but the reverse is not true. In particular, the empty set cannot be pointed, because it has no element that can be chosen as the basepoint.
Article summary
In mathematics, a pointed set (also based set or rooted set ) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point (also spelled basepoint). A map between pointed a sets (X, x_0) and (Y, y_0) —called a based map, pointed map, or point-preserving map —is a function from X to Y that maps one basepoint to another, i.e. a map f : X \to Y such that . A based map is usually denoted . Pointed sets are very simple algebraic structures. In the sense of universal algebra, a pointed set is a set X together with a single nullary operation , which picks out the basepoint. Pointed maps are the homomorphisms of these algebraic structures. The class of all pointed sets together with the class of all based maps forms a category. Every pointed set can be converted to an ordinary set by forgetting the basepoint (the forgetful functor is faithful), but the reverse is not true. In particular, the empty set cannot be pointed, because it has no element that can be chosen as the basepoint.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.