# Specialization of a point

$x$ in a topological space $X$

A point $y \in X$ for which the inclusion $y \in \overline{\{x\}}$ holds; equivalently the inclusion $\overline{\{y\}} \subseteq \overline{\{x\}}$; every neighbourhood of $x$ is a neighbourhood of $y$.

A point $x$ is called generic if any point of $X$ is a specialization of it, that is, if $\overline{\{x\}} = X$. The other extreme case is that of a closed point: a point which has a unique specialization, namely the point itself.

For the affine scheme $\mathrm{Spec}(A)$ of a ring $A$, a point $y$ is a specialization of a point $x$ if for the corresponding prime ideals of $A$ the inclusion $\mathfrak{p}_x \subseteq \mathfrak{p}_y$ holds. When $A$ is a ring without zero divisors, the point $\{0\}$ is the generic one. The relation of specialization distributes into levels: the highest are the closed points, on the next level are the points whose specializations are closed, and on the $i$-th level are the points whose specializations belong to the levels with labels $\le i-1$. For example, for $\mathrm{Spec}(\mathbf{C}[T_1,\ldots,T_n]$ there are $n+1$ levels: closed points, generic points of curves, generic points of surfaces,$\ldots$, the generic point of the $n$-dimensional affine space.

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How to Cite This Entry:
Specialization of a point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Specialization_of_a_point&oldid=37226
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article