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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s0863001.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s0863002.png" />''
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''$x$ in a topological space $X$''
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s0863003.png" /> for which the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s0863004.png" /> holds (this is equivalent to the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s0863005.png" />). A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s0863006.png" /> is called generic if any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s0863007.png" /> is a specialization of it, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s0863008.png" />. The other extreme case is that of a closed point: A point which has a unique specialization, namely the point itself.
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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 $y$.
  
For the [[Affine scheme|affine scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s0863009.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630010.png" />, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630011.png" /> is a specialization of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630012.png" /> if for the corresponding prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630013.png" /> the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630014.png" /> holds. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630015.png" /> is a ring without zero divisors, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630016.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630017.png" />-th level are the points whose specializations belong to the levels with labels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630018.png" />. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630019.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630020.png" /> levels: closed points, generic points of curves, generic points of surfaces<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630021.png" /> the generic point of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630022.png" />-dimensional affine space.
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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.
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.I. Manin, "Lectures on algebraic geometry" , '''1''' , Moscow (1970) (In Russian) {{MR|0284434}} {{ZBL|0204.21302}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" , '''I. Le langage des schémes''' , Springer (1971) {{MR|0217085}} {ZBL|0203.23301}} </TD></TR></table>
 
  
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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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====Comments====
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Here $\overline{\{x\}}$ denotes the [[Closure of a set|closure]] of the [[singleton]] set $\{x\}$. The closure of a point is an [[Irreducible topological space|irreducible]] subset of $X$, and conversely, every irreducible subset of $X$ has a generic point.
  
====Comments====
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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]].
Of course, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630023.png" /> denotes the closure of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630024.png" />. The closure of a point is an irreducible subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630025.png" />, and conversely, every irreducible subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086300/s08630026.png" /> has a generic point.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.I. Manin, "Lectures on algebraic geometry" , '''1''' , Moscow (1970) (In Russian) {{MR|0284434}} {{ZBL|0204.21302}} </TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique" , '''I. Le langage des schémas''' , Springer (1971) {{MR|0217085}} {{ZBL|0203.23301}} </TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR>
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<TR><TD valign="top">[b1]</TD> <TD valign="top"> Steven Vickers ''Topology via Logic'' Cambridge Tracts in Theoretical Computer Science '''5''' Cambridge University Press (1989) {{ISBN|0-521-36062-5}} {{ZBL|0668.54001}} </TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 05:39, 18 July 2024

$x$ 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 géométrie algébrique" , I. Le langage des schémas , 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
How to Cite This Entry:
Specialization of a point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Specialization_of_a_point&oldid=23981
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article