Sober space

Last updated November 17, 2024

In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every nonempty irreducible closed subset has a unique generic point.

Definitions

Sober spaces have a variety of cryptomorphic definitions, which are documented in this section ^[1]^[2]. In each case below, replacing "unique" with "at most one" gives an equivalent formulation of the T₀ axiom. Replacing it with "at least one" is equivalent to the property that the T₀ quotient of the space is sober, which is sometimes referred to as having "enough points" in the literature.

With irreducible closed sets

A closed set is irreducible if it cannot be written as the union of two proper closed subsets. A space is sober if every nonempty irreducible closed subset is the closure of a unique point.

In terms of morphisms of frames and locales

A topological space X is sober if every map that preserves all joins and all finite meets from its partially ordered set of open subsets to $\{0,1\}$ is the inverse image of a unique continuous function from the one-point space to X.

This may be viewed as a correspondence between the notion of a point in a locale and a point in a topological space, which is the motivating definition.

Using completely prime filters

A filter F of open sets is said to be completely prime if for any family $O_{i}$ of open sets such that $\bigcup _{i}O_{i}\in F$ , we have that $O_{i}\in F$ for some i. A space X is sober if each completely prime filter is the neighbourhood filter of a unique point in X.

In terms of nets

A net $x_{\bullet }$ is self-convergent if it converges to every point $x_{i}$ in $x_{\bullet }$ , or equivalently if its eventuality filter is completely prime. A net $x_{\bullet }$ that converges to $x$ converges strongly if it can only converge to points in the closure of $x$ . A space is sober if every self-convergent net $x_{\bullet }$ converges strongly to a unique point $x$ .^[2]

In particular, a space is T1 and sober precisely if every self-convergent net is constant.

As a property of sheaves on the space

A space X is sober if every functor from the category of sheaves Sh(X) to Set that preserves all finite limits and all small colimits must be the stalk functor of a unique point x.

Properties and examples

Any Hausdorff (T₂) space is sober (the only irreducible subsets being points), and all sober spaces are Kolmogorov (T₀), and both implications are strict.^[3]

Sobriety is not comparable to the T₁ condition:

an example of a T₁ space which is not sober is an infinite set with the cofinite topology, the whole space being an irreducible closed subset with no generic point;
an example of a sober space which is not T₁ is the Sierpinski space.

Moreover T₂ is stronger than T₁and sober, i.e., while every T₂ space is at once T₁ and sober, there exist spaces that are simultaneously T₁ and sober, but not T₂. One such example is the following: let X be the set of real numbers, with a new point p adjoined; the open sets being all real open sets, and all cofinite sets containing p.

Sobriety of X is precisely a condition that forces the lattice of open subsets of X to determine X up to homeomorphism, which is relevant to pointless topology.

Sobriety makes the specialization preorder a directed complete partial order.

Every continuous directed complete poset equipped with the Scott topology is sober.

Finite T₀ spaces are sober.^[4]

The prime spectrum Spec(R) of a commutative ring R with the Zariski topology is a compact sober space.^[3] In fact, every spectral space (i.e. a compact sober space for which the collection of compact open subsets is closed under finite intersections and forms a base for the topology) is homeomorphic to Spec(R) for some commutative ring R. This is a theorem of Melvin Hochster.^[5] More generally, the underlying topological space of any scheme is a sober space.

The subset of Spec(R) consisting only of the maximal ideals, where R is a commutative ring, is not sober in general.

Related Research Articles

<span class="mw-page-title-main">Compact space</span> Type of mathematical space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers $is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers is not compact either, because it excludes the two limiting values and . However, the extended real number line would be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.$

In topology and related branches of mathematics, a Hausdorff space ( HOWSS-dorf, HOWZ-dorf), T₂ space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T₂) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.

In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology. For a list of terms specific to algebraic topology, see Glossary of algebraic topology.

In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.

In topology and related branches of mathematics, a topological space X is a T₀ space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other. In a T₀ space, all points are topologically distinguishable.

In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1935 article by Tychonoff, "Über einen Funktionenraum".

In topology and related branches of mathematics, a T₁ space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R₀ space is one in which this holds for every pair of topologically distinguishable points. The properties T₁ and R₀ are examples of separation axioms.

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.

In mathematics, a cofinite subset of a set $is a subset whose complement in is a finite set. In other words, contains all but finitely many elements of If the complement is not finite, but is countable, then one says the set is cocountable.$

In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński.

In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the T₀ separation axiom, this preorder is even a partial order (called the specialization order). On the other hand, for T₁ spaces the order becomes trivial and is of little interest.

In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topoi.

The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable. It follows that the only closed subsets are X and the countable subsets of X. Symbolically, one writes the topology as

In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component of an algebraic set is an algebraic subset that is irreducible and maximal for this property. For example, the set of solutions of the equation $xy = 0$ is not irreducible, and its irreducible components are the two lines of equations $x = 0$ and $y = 0$ .

In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space X that cannot be written as the union of two proper closed subsets. The name irreducible space is preferred in algebraic geometry.

In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.

In mathematics, a scattered space is a topological space X that contains no nonempty dense-in-itself subset. Equivalently, every nonempty subset A of X contains a point isolated in A.

References

↑ Mac Lane, Saunders (1992). Sheaves in geometry and logic: a first introduction to topos theory. New York: Springer-Verlag. pp. 472–482. ISBN 978-0-387-97710-2.
1 2 Sünderhauf, Philipp (1 December 2000). "Sobriety in Terms of Nets". Applied Categorical Structures. 8 (4): 649–653. doi:10.1023/A:1008673321209.
1 2 Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004). Encyclopedia of general topology . Elsevier. pp. 155–156. ISBN 978-0-444-50355-8.
↑ "General topology - Finite $T_0$ spaces are sober".
↑ Hochster, Melvin (1969), "Prime ideal structure in commutative rings", Trans. Amer. Math. Soc., 142: 43–60, doi: 10.1090/s0002-9947-1969-0251026-x