T1 space

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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related branches of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a T1 space is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
in which, for every pair of distinct points, each has a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural ...
not containing the other point. An R0 space is one in which this holds for every pair of
topologically distinguishable In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
points. The properties T1 and R0 are examples of separation axioms.

# Definitions

Let ''X'' be a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
and let ''x'' and ''y'' be points in ''X''. We say that ''x'' and ''y'' are if each lies in a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
that does not contain the other point. * ''X'' is called a T1 space if any two distinct points in ''X'' are separated. * ''X'' is called an R0 space if any two
topologically distinguishable In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
points in ''X'' are separated. A T1 space is also called an accessible space or a space with Fréchet topology and an R0 space is also called a symmetric space. (The term also has an entirely different meaning in
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 (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
. 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 In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
. The term also has another meaning.) A topological space is a T1 space if and only if it is both an R0 space and a Kolmogorov (or T0) space (i.e., a space in which distinct points are topologically distinguishable). A topological space is an R0 space if and only if its
Kolmogorov quotient 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 th ...
is a T1 space.

# Properties

If $X$ is a topological space then the following conditions are equivalent: #$X$ is a T1 space. #$X$ is a T0 space and an R0 space. #Points are closed in $X$; that is, for every point $x \in X,$ the singleton set $\$ is a
closed subset 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 ...
of $X.$ #Every subset of $X$ is the intersection of all the open sets containing it. #Every
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. ...
is closed. #Every cofinite set of $X$ is open. #For every $x \in X,$ the fixed ultrafilter at $x$ converges only to $x.$ #For every subset $S$ of $X$ and every point $x \in X,$ $x$ is a
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point 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 cont ...
of $S$ if and only if every open
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of $x$ contains infinitely many points of $S.$ #Each map from the Sierpinski space to $X$ is trivial. # The map from the Sierpinski space to the single point has the
lifting property In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given c ...
with respect to the map from $X$ to the single point. If $X$ is a topological space then the following conditions are equivalent: (where $\operatorname\$ denotes the closure of $\$) #$X$ is an R0 space. #Given any $x \in X,$ the closure of $\$ contains only the points that are topologically indistinguishable from $x.$ #The
Kolmogorov quotient 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 th ...
of $X$ is T1. #For any $x,y\in X,$ $x$ is in the closure of $\$ if and only if $y$ is in the closure of $\.$ #The specialization preorder on $X$ is
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
(and therefore an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
). #The sets $\operatorname\$ for $x\in X$ form a partition of $X$ (that is, any two such sets are either identical or disjoint). #If $F$ is a closed set and $x$ is a point not in $F$, then $F\cap\operatorname\=\emptyset.$ #Every
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of a point $x\in X$ contains $\operatorname\.$ #Every
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...
is a union of closed sets. #For every $x \in X,$ the fixed ultrafilter at $x$ converges only to the points that are topologically indistinguishable from $x.$ In any topological space we have, as properties of any two points, the following implications : $\implies$ $\implies$ 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. A space is T1 if and only if it is both R0 and T0. A finite T1 space is necessarily discrete (since every set is closed).

# Examples

* Sierpinski space is a simple example of a topology that is T0 but is not T1, and hence also not R0. * The overlapping interval topology is a simple example of a topology that is T0 but is not T1. * Every weakly Hausdorff space is T1 but the converse is not true in general. * The cofinite topology on an infinite set is a simple example of a topology that is T1 but is not Hausdorff (T2). This follows since no two open sets of the cofinite topology are disjoint. Specifically, let $X$ be the set of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, and define the
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...
s $O_A$ to be those subsets of $X$ that contain all but a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
subset $A$ of $X.$ Then given distinct integers $x$ and $y$: :* the open set $O_$ contains $y$ but not $x,$ and the open set $O_$ contains $x$ and not $y$; :* equivalently, every singleton set $\$ is the complement of the open set $O_,$ 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 intersection of any two open sets $O_A$ and $O_B$ is $O_A \cap O_B = O_,$ which is never empty. Alternatively, the set of even integers is compact but not closed, 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 $O_A$ from the previous example, define a
subbase In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
of open sets $G_x$ for any integer $x$ to be $G_x = O_$ if $x$ is an
even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 4 ...
, and $G_x = O_$ if $x$ is odd. Then the basis of the topology are given by finite intersections of the subbasic sets: given a finite set $A,$the open sets of $X$ are ::$U_A := \bigcap_ G_x.$ :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 topology on an algebraic variety (over an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
) is T1. To see this, note that the singleton containing a point with
local coordinates Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples: * Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow ...
$\left\left(c_1, \ldots, c_n\right\right)$ is the zero set of the
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
s $x_1 - c_1, \ldots, x_n - c_n.$ Thus, the point is closed. However, this example is well known as a space that is not Hausdorff (T2). The Zariski topology is essentially an example of a cofinite topology. * The Zariski topology on a
commutative ring In mathematics, 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 ring properties that are not ...
(that is, the prime
spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
) is T0 but not, in general, T1.Arkhangel'skii (1990). ''See example 21, section 2.6.'' To see this, note that the closure of a one-point set is the set of all
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
s that contain the point (and thus the topology is T0). However, this closure is a maximal ideal, and the only closed points are the maximal ideals, and are thus not contained in any of the open sets of the topology, and thus the space does not satisfy axiom T1. To be clear about this example: the Zariski topology for a commutative ring $A$ is given as follows: the topological space is the set $X$ of all
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
s of $A.$ The base of the topology is given by the open sets $O_a$ of prime ideals that do contain $a \in A.$ It is straightforward to verify that this indeed forms the basis: so $O_a \cap O_b = O_$ and $O_0 = \varnothing$ and $O_1 = X.$ The closed sets of the Zariski topology are the sets of prime ideals that contain $a.$ Notice how this example differs subtly from the cofinite topology example, above: the points in the topology are not closed, in general, whereas in a T1 space, points are always closed. * Every
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set ...
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 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 that is used to define uniform properties such as completeness, uniform continuity and un ...
s,
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 deriv ...
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 spaces.