Difference between revisions of "Generic point"
From Encyclopedia of Mathematics
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A point in a topological space whose closure coincides with the whole space. A topological space having a generic point is an [[Irreducible topological space|irreducible topological space]]; however, an irreducible space may have no generic point or may have many generic points. However, if the space satisfies the [[Kolmogorov axiom|Kolmogorov axiom]], then it can have at most one generic point. Any irreducible algebraic variety or irreducible scheme has a unique generic point. In this case the generic point is just the spectrum of the field of rational functions on the variety. | A point in a topological space whose closure coincides with the whole space. A topological space having a generic point is an [[Irreducible topological space|irreducible topological space]]; however, an irreducible space may have no generic point or may have many generic points. However, if the space satisfies the [[Kolmogorov axiom|Kolmogorov axiom]], then it can have at most one generic point. Any irreducible algebraic variety or irreducible scheme has a unique generic point. In this case the generic point is just the spectrum of the field of rational functions on the variety. | ||
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| − | A space satisfying the Kolmogorov axiom is usually called a | + | A space satisfying the Kolmogorov axiom is usually called a $T_0$-space. |
Latest revision as of 14:57, 1 May 2014
A point in a topological space whose closure coincides with the whole space. A topological space having a generic point is an irreducible topological space; however, an irreducible space may have no generic point or may have many generic points. However, if the space satisfies the Kolmogorov axiom, then it can have at most one generic point. Any irreducible algebraic variety or irreducible scheme has a unique generic point. In this case the generic point is just the spectrum of the field of rational functions on the variety.
The term "generic point" is sometimes also used to denote a point in general position.
Comments
A space satisfying the Kolmogorov axiom is usually called a $T_0$-space.
How to Cite This Entry:
Generic point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generic_point&oldid=13406
Generic point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generic_point&oldid=13406
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article