Sober room

A sober space is a space considered in the mathematical theory of topological spaces , which is characterized by the fact that its closed, irreducible sets (see below) are easy to describe. The term fasting ( Engl. Sober) goes to M. Artin , A. Grothendieck and J.Verdier back.

Terms

A non-empty , closed set of a topological space is called irreducible if it is not a union of two real, closed subsets, that is, is with two closed subsets , then must or be. ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A=A_{1}\cup A_{2}}$ ${\displaystyle A_{1},A_{2}\subset A}$ ${\displaystyle A=A_{1}}$ ${\displaystyle A=A_{2}}$

An example is the end of a point , because if like above, one of the sets must contain the point and thus also its end, that is, it follows . In general, irreducible sets are not of this form, and if they are of this form then the point need not necessarily be unique. This motivates the following definition: ${\displaystyle A={\overline {\{x\}}}}$ ${\displaystyle x\in X}$ ${\displaystyle A=A_{1}\cup A_{2}}$ ${\displaystyle A_{i}}$ ${\displaystyle x}$ ${\displaystyle A=A_{i}}$ ${\displaystyle x}$

definition

A topological space is called sober if every closed, irreducible subset is the closure of exactly one point.

That means more precisely: A topological space is called sober if every closed, irreducible subset is of the form with a clearly defined point . ${\displaystyle X}$ ${\displaystyle A\subset X}$ ${\displaystyle A={\overline {\{x\}}}}$ ${\displaystyle x\in X}$

The uniquely determined point with is called the generic point of . ${\displaystyle x}$ ${\displaystyle A={\overline {\{x\}}}}$ ${\displaystyle A}$

Examples

• Every Hausdorff space is sober, because the closed, irreducible subsets are exactly the one-element subsets.
• The two-element space with the open sets and is sober, because and are the only closed, irreducible sets. This is therefore an example of a sober room that is not Hausdorff-like, because it is not even a T ₁ room . ${\displaystyle X=\{x,y\}}$ ${\displaystyle \emptyset ,X}$ ${\displaystyle \{x\}}$ ${\displaystyle X={\overline {\{x\}}}}$ ${\displaystyle \{y\}={\overline {\{y\}}}}$
• The topological space with the cofinite topology is a T ₁ space that is not sober. Since, in addition to the total space, only the finite sets are closed, the total space is closed and irreducible but not equal to the end of a point, that is, it is not sober. ${\displaystyle X=\mathbb {N} }$ ${\displaystyle X}$
• The spectrum of a commutative ring with a single element is sober with the Zariski topology . Conversely, every quasi-compact , sober room is of this shape. ${\displaystyle X=\mathrm {Spec} (R)}$

properties

T ₀ property and alternative definitions

Sober spaces are T ₀ -spaces , because for every two different points and because of the uniqueness condition of the above definition , that is, it is or , from which one can easily get the T ₀ -property. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle {\overline {\{x\}}}\not ={\overline {\{y\}}}}$ ${\displaystyle x\notin {\overline {\{y\}}}}$ ${\displaystyle y\notin {\overline {\{x\}}}}$

Some authors dispense with the uniqueness condition in the definition of a sober space and instead demand the T ₀ property. This turns out to be equivalent, since the uniqueness condition follows from the T ₀ property.

Another alternative definition is obtained by going over to complements and then formulating the definition condition using open sets:

A topological space is sober when it for any real, open subset with the property that for all open sets out already or , a clear follow exists so that applies. ${\displaystyle X}$ ${\displaystyle P\subset X}$ ${\displaystyle U,V\subset X}$ ${\displaystyle U\cap V\subset P}$ ${\displaystyle U\subset P}$ ${\displaystyle V\subset P}$ ${\displaystyle x\in X}$ ${\displaystyle P=X\setminus {\overline {\{x\}}}}$

Classification in the axioms of separation

Since sober spaces are T ₀ according to the above , sobriety like T _{1 is} a property between T ₀ and T ₂ (Hausdorff property). T ₁ and sobriety do not allow direct comparability, because according to the above examples there are rooms that have one of the properties but not the other. However, as the following categorical characteristics show, sobriety is more of a seclusion than a separation characteristic .

Categorical Properties and Sobrification

Let it be the sub-category of sober spaces in the category of all topological spaces. Then the forget functor has a left adjoint functor , which is called "sobrification" in English-language literature, which can be translated as creating sobriety. ${\displaystyle {\mathcal {Sob}}}$ ${\displaystyle {\mathcal {T\!\!op}}}$ ${\displaystyle V:{\mathcal {Sob}}\rightarrow {\mathcal {T\!\!op}}}$ ${\displaystyle S:{\mathcal {T\!\!op}}\rightarrow {\mathcal {Sob}}}$

The construction of the functor looks like this. If an arbitrary, topological space is, then be the set of all irreducible, closed subsets. For every open set is ${\displaystyle S}$ ${\displaystyle X}$ ${\displaystyle S(X)}$ ${\displaystyle U\subset X}$

${\displaystyle U^{S}:=\{A\in S(X)\mid A\cap U\not =\emptyset \}\subset S(X)}$.

Then they form the open sets of a topology that makes a sober space. The canonical figure ${\displaystyle U^{S}}$ ${\displaystyle S(X)}$

${\displaystyle j_{X}:X\rightarrow S(X),\quad x\mapsto {\overline {\{x\}}}}$

is steady. Is continuous, so be ${\displaystyle f:X\rightarrow Y}$

${\displaystyle S(f):S(X)\rightarrow S(Y),\quad A\mapsto {\overline {f(A)}}}$.

These definitions make the above sobrification functor. ${\displaystyle S}$

The left adjointness to the forget function mentioned above means the following universal property : If a topological space and a continuous mapping, where a sober space is, then there is exactly one continuous mapping such that . ${\displaystyle V}$ ${\displaystyle X}$ ${\displaystyle f:X\rightarrow V(Y)}$ ${\displaystyle Y}$ ${\displaystyle {\overline {f}}:S(X)\rightarrow Y}$ ${\displaystyle f=V({\overline {f}})\circ j_{X}}$

If sober, then there is a homeomorphism , that is, the transition to brings nothing new. In this sense, the use of the functor is a final illustration and sober rooms can be viewed as the closed rooms. ${\displaystyle X}$ ${\displaystyle j_{X}:X\rightarrow S(X)}$ ${\displaystyle S(X)}$ ${\displaystyle S}$ ${\displaystyle S}$

Individual evidence

1. ↑ M. Artin, A. Grothendieck, J. Verdier: Séminaire de Géométrie Algébrique du Bois-Marie , 1963–1964
2. ↑ CE Aull, R. Lowen: Handbook of the History of General Topology , Volume 1, Kluwer Academic Publishers 1997, ISBN 0-7923-4479-0 , page 325
3. ↑ Michel Marie Deza, Elena Deza: Encyclopedia of Distances , 2nd edition, Springer Verlag, ISBN 978-3-642-30957-1 , page 62.
4. ↑ M. Hochster: Prime ideal structure in commutative rings , Trans. Amer. Math. Soc. 142, (1969), pp. 43-60. (Here the corresponding spaces are called spectral).
5. ↑ Jean Goubault-Larrecq: Non-Hausdorff Topology and Domain Theory: Selected Topics in Point-Set Topology , Cambridge University Press 2013, ISBN 978-1-107-03413-6 , Chapter 8.2 Sober spaces and sobrification , Definition 8.2.4
6. ^ S. Mac Lane, I. Moerdijk: Sheaves in Geometry and Logic , Springer-Verlag 1992, ISBN 0-387-97710-4 , Chapter IX, Paragraph 2, Definition 2
7. ^ PT Johnstone: Topos Theory , Academic Press 1977, ISBN 0-12-387850-0 , sentence 7.22