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.

Comments

Here $\overline{\{x\}}$ denotes the closure of the singleton set $\{x\}$. The closure of a point is an irreducible subset of $X$, and conversely, every irreducible subset of $X$ has a generic point.

The relation "$y$ is a specialisation of $x$" on $X$, denoted $y \sqsupseteq x$, is reflexive and transitive. It is anti-symmetric, and hence a partial order on $X$, if and only if, $X$ is a T0 space.

References

[1]	Yu.I. Manin, "Lectures on algebraic geometry" , 1 , Moscow (1970) (In Russian) MR0284434 Zbl 0204.21302
[2]	A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" , I. Le langage des schémes , Springer (1971) MR0217085 Zbl 0203.23301
[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001
[b1]	Steven Vickers Topology via Logic Cambridge Tracts in Theoretical Computer Science 5 Cambridge University Press (1989) ISBN 0-521-36062-5 Zbl 0668.54001