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 irreducible closed subset has a unique generic point.

All Hausdorff (T_{2}) spaces are sober (the only irreducible subsets being points), and all sober spaces are Kolmogorov (T_{0}) spaces. Both implications are strict.^{ [1] }

T_{1} spaces need not be sober. An example of a T_{1} 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. Conversely, the Sierpinski space is an example of a sober spaces which is not a T_{1} space.

All T_{2} spaces are necessarily both T_{1} and sober, but there exist spaces that are simultaneously T_{1} and sober, but not T_{2}. 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.

The prime spectrum Spec(*R*) of a commutative ring *R* with the Zariski topology is a compact sober space.^{ [1] } 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.^{ [2] } 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.

- Stone duality, on the duality between topological spaces which are sober and frames (i.e. complete Heyting algebras) which are spatial.

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In topology and related branches of mathematics, a **Hausdorff space**, **separated space** or **T _{2} space** is a topological space where for any two distinct points there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T

In mathematics, a **topological space** is, roughly speaking, a geometrical space in which *closeness* is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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.

In commutative algebra the **prime spectrum** of a ring like *R* is the set of all prime ideals of *R* which is usually denoted by , in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings .

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

In ring theory, a branch of abstract algebra, a **commutative ring** is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

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 1937 paper of Eduard Čech.

In topology and related branches of mathematics, a **T _{1} space** is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An

In algebraic geometry and commutative algebra, the **Zariski topology** is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis, particularly it is not Hausdorff. This topology introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring.

In mathematics, a **scheme** is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.

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

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 T_{0} separation axiom, this preorder is even a partial order (called the **specialization order**). On the other hand, for T_{1} 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 topos.

In algebraic geometry, a **proper morphism** between schemes is an analog of a proper map between complex analytic spaces.

In mathematics, the **spectrum of a C*-algebra** or **dual of a C*-algebra***A*, denoted *Â*, is the set of unitary equivalence classes of irreducible *-representations of *A*. A *-representation π of *A* on a Hilbert space *H* is **irreducible** if, and only if, there is no closed subspace *K* different from *H* and {0} which is invariant under all operators π(*x*) with *x* ∈ *A*. We implicitly assume that irreducible representation means *non-null* irreducible representation, thus excluding trivial representations on one-dimensional spaces. As explained below, the spectrum *Â* is also naturally a topological space; this is similar to the notion of the spectrum of a ring.

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** 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 mathematics, a **Noetherian topological space**, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong compactness condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that *every* subset is compact.

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 sets. The name *irreducible space* is preferred in algebraic geometry.

- 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. - ↑ Hochster, Melvin (1969), "Prime ideal structure in commutative rings",
*Trans. Amer. Math. Soc.*,**142**: 43–60, doi: 10.1090/s0002-9947-1969-0251026-x

- Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004).
*Categorical foundations. Special topics in order, topology, algebra, and sheaf theory*. Encyclopedia of Mathematics and Its Applications.**97**. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001. - Vickers, Steven (1989).
*Topology via logic*. Cambridge Tracts in Theoretical Computer Science.**5**. Cambridge: Cambridge University Press. p. 66. ISBN 0-521-36062-5. Zbl 0668.54001.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.