Sober space
Encyclopedia
In mathematics
, a sober space is a topological space
such that every irreducible
closed subset of X is the closure
of exactly one point of X: that is, has a unique generic point
.
sober spaces are Kolmogorov (T0). Sobriety is
not comparable
to the T1
condition.
Moreover T2 is stronger than T1 and 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.
The prime spectrum Spec(R) of a commutative ring
R with the Zariski topology
is a compact
sober space. 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
.
More generally, the underlying topological space of any scheme
is a sober space.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a sober space is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
such that every irreducible
Hyperconnected space
In mathematics, a hyperconnected 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....
closed subset of X is the closure
Closure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
of exactly one point of X: that is, has a unique generic point
Generic point
In mathematics, in the fields general topology and particularly of algebraic geometry, a generic point P of a topological space X is an algebraic way of capturing the notion of a generic property: a generic property is a property of the generic point.- Definition and motivation :A generic point of...
.
Properties and examples
Any Hausdorff (T2) space is sober (the only irreducible subsets being points), and allsober spaces are Kolmogorov (T0). Sobriety is
not comparable
Comparability
In mathematics, any two elements x and y of a set P that is partially ordered by a binary relation ≤ are comparable when either x ≤ y or y ≤ x...
to the T1
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
condition.
Moreover T2 is stronger than T1 and 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
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
, which is relevant to pointless topology
Pointless topology
In mathematics, pointless topology is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann...
.
Sobriety makes the specialization preorder a directed complete partial order.
The prime spectrum Spec(R) of a commutative ring
Commutative ring
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....
R with the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
is a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
sober space. In fact, every spectral space
Spectral space
In mathematics, a spectral space is a topological space which is homeomorphic to the spectrum of a commutative ring.-Definition:Let X be a topological space and let K\circ be the set of allquasi-compact and open subsets of X...
(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
Melvin Hochster
Melvin Hochster is an eminent American mathematician, regarded as one of the leading commutative algebraists active today. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan.Hochster attended Stuyvesant High School, where he was...
.
More generally, the underlying topological space of any scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
is a sober space.
See also
- Stone dualityStone dualityIn mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation...
, on the duality between topological spaces which are sober and frames (i.e. complete Heyting algebraComplete Heyting algebraIn mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames...
s) which are spatial.
External links
- Discussion of weak separation axioms (PDF file)