Closed point

The closed point is a term used in set theoretic topology , which is particularly important in algebraic geometry .

definition

A closed point in a topological space is a point such that the one-element subset is a closed subset of . ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle \left\{x\right\}}$ ${\displaystyle X}$

Closed points in algebraic geometry

In the Zariski topology of an algebraic variety , the closed points correspond to the maximum ideals of . ${\displaystyle \operatorname {Spec} (R)}$ ${\displaystyle R}$

For example, the prime ideals different from the zero ideal , that is, the main ideals generated by the prime numbers , correspond to the closed points in . The zero ideal is also a prime ideal, but not a closed point. ${\displaystyle \{0\}}$ ${\displaystyle \operatorname {Spec} (\mathbb {Z} )}$

T1 rooms

A topological space is a T ₁ -space if and only if all points are closed points.

Web links

• closed point (nLab)