T1 space
Encyclopedia
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. The properties T1 and R0 are examples of separation axiom
s.
and let x and y be points in X. We say that x and y can be separated
if each lies in an open set
which does not contain the other point.
A T1 space is also called an accessible space or a Fréchet space and a R0 space is also called a symmetric space. (The term Fréchet space also has an entirely different meaning
in functional analysis
. For this reason, the term T1 space is preferred. There is also a notion of a Fréchet-Urysohn space as a type of sequential space
. The term symmetric space has another meaning
.)
Let X be a topological space. Then the following conditions are equivalent:
In any topological space we have, as properties of any two points, the following implications
If the first arrow can be reversed the space is R0. If the second arrow can be reversed the space is T0. If the composite arrow can be reversed the space is T1. Clearly, a space is T1 if and only if it's both R0 and T0.
Note that a finite T1 space is necessarily discrete
(since every set is closed).
s, Cauchy space
s, and convergence spaces.
The characteristic that unites the concept in all of these examples is that limits of fixed ultrafilters (or constant nets) are unique (for T1 spaces) or unique up to topological indistinguishability (for R0 spaces).
As it turns out, uniform spaces, and more generally Cauchy spaces, are always R0, so the T1 condition in these cases reduces to the T0 condition.
But R0 alone can be an interesting condition on other sorts of convergence spaces, such as pretopological space
s.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and related branches of mathematics
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 T1 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...
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. The properties T1 and R0 are examples of separation axiom
Separation axiom
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...
s.
Definitions
Let X be a topological spaceTopological 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...
and let x and y be points in X. We say that x and y can be separated
Separated sets
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way....
if each lies in an open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
which does not contain the other point.
- X is a T1 space if any two distinct points in X can be separated.
- X is a R0 space if any two topologically distinguishable points in X can be separated.
A T1 space is also called an accessible space or a Fréchet space and a R0 space is also called a symmetric space. (The term Fréchet space also has an entirely different meaning
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces...
in functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
. For this reason, the term T1 space is preferred. There is also a notion of a Fréchet-Urysohn space as a type of sequential space
Sequential space
In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology....
. The term symmetric space has another meaning
Symmetric space
A symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point...
.)
Properties
Let X be a topological space. Then the following conditions are equivalent:- X is a T1 space.
- X is a T0 spaceKolmogorov spaceIn topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other. This condition, called the T0 condition, is one of the separation axioms...
and a R0 space. - Points are closed in X; i.e. given any x in X, the singleton set {x} is a closed setClosed setIn 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...
. - Every subset of X is the intersection of all the open sets containing it.
- Every finite set is closed.
- Every cofiniteCofiniteIn 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...
set of X is open. - The fixed ultrafilter at x converges only to x.
- For every point x in X and every subset S of X, x is a limit pointLimit pointIn mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
of S if and only if every open neighbourhood of x contains infinitely many points of S.
Let X be a topological space. Then the following conditions are equivalent:
- X is an R0 space.
- Given any x in X, the closureClosure (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 {x} contains only the points that x is topologically indistinguishable from. - The specialization preorder on X is symmetricSymmetric relationIn mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...
(and therefore an equivalence relationEquivalence relationIn mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
). - The fixed ultrafilter at x converges only to the points that x is topologically indistinguishable from.
- The Kolmogorov quotient of X (which identifies topologically indistinguishable points) is T1.
- Every open setOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
is the union of closed sets.
In any topological space we have, as properties of any two points, the following implications
- separated ⇒ topologically distinguishable ⇒ distinct
If the first arrow can be reversed the space is R0. If the second arrow can be reversed the space is T0. If the composite arrow can be reversed the space is T1. Clearly, a space is T1 if and only if it's both R0 and T0.
Note that a finite T1 space is necessarily discrete
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
(since every set is closed).
Examples
- Sierpinski spaceSierpinski spaceIn 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...
is a simple example of a topology that is T0 but is not T1. - The overlapping interval topologyOverlapping interval topologyIn mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.-Definition:Given the closed interval [-1,1] of the real number line, the open sets of the topology are generated from the half-open intervals [-1,b) and In mathematics, the...
is a simple example of a topology that is T0 but is not T1.
- The cofinite topology on an infinite set is a simple example of a topology that is T1 but is not HausdorffHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
(T2). This follows since no two open sets of the cofinite topology are disjoint. Specifically, let X be the set of integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, and define the open setOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s OA to be those subsets of X which contain all but a finite subset A of X. Then given distinct integers x and y: - the open set O{x} contains y but not x, and the open set O{y} contains x and not y;
- equivalently, every singleton set {x} is the complement of the open set O{x}, so it is a closed set;
- so the resulting space is T1 by each of the definitions above. This space is not T2, because the intersectionIntersection (set theory)In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of any two open sets OA and OB is OA∪B, which is never empty. Alternatively, the set of even integers is compact but not closedClosed setIn 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...
, which would be impossible in a Hausdorff space.
- The above example can be modified slightly to create the double-pointed cofinite topology, which is an example of an R0 space that is neither T1 nor R1. Let X be the set of integers again, and using the definition of OA from the previous example, define a subbaseSubbaseIn topology, a subbase for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B...
of open sets Gx for any integer x to be Gx = O{x, x+1} if x is an even number, and Gx = O{x-1, x} if x is odd. Then the basis of the topology are given by finite intersectionsIntersection (set theory)In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of the subbasis sets: given a finite set A, the open sets of X are
- The resulting space is not T0 (and hence not T1), because the points x and x + 1 (for x even) are topologically indistinguishable; but otherwise it is essentially equivalent to the previous example.
- The Zariski topologyZariski topologyIn 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...
on an algebraic varietyAlgebraic varietyIn mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
(over an algebraically closed fieldAlgebraically closed fieldIn mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
) is T1. To see this, note that a point with local coordinatesLocal coordinatesLocal coordinates are measurement indices into a local coordinate system or a local coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc.Local...
(c1,...,cn) is the zero set of the polynomialPolynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s x1-c1, ..., xn-cn. Thus, the point is closed. However, this example is well known as a space that is not HausdorffHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
(T2). The Zariski topology is essentially an example of a cofinite topology.
- Every totally disconnected space is T1, since every point is a connected component and therefore closed.
Generalisations to other kinds of spaces
The terms "T1", "R0", and their synonyms can also be applied to such variations of topological spaces as uniform spaceUniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
s, Cauchy space
Cauchy space
In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to...
s, and convergence spaces.
The characteristic that unites the concept in all of these examples is that limits of fixed ultrafilters (or constant nets) are unique (for T1 spaces) or unique up to topological indistinguishability (for R0 spaces).
As it turns out, uniform spaces, and more generally Cauchy spaces, are always R0, so the T1 condition in these cases reduces to the T0 condition.
But R0 alone can be an interesting condition on other sorts of convergence spaces, such as pretopological space
Pretopological space
In general topology, a pretopological space is a generalization of the concept of topological space. A pretopological space can be defined as in terms of either filters or a preclosure operator.Let X be a set...
s.