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 a neighborhood not containing the other point.^[1] 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.

Let X be a topological space and let x and y be points in X. We say that x and y are separated if each lies in a neighbourhood 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 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 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 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 is a T₁ space.

If ${\displaystyle X}$ is a topological space then the following conditions are equivalent:

1. ${\displaystyle X}$ is a T₁ space.
2. ${\displaystyle X}$ is a T₀ space and an R₀ space.
3. Points are closed in ${\displaystyle X}$; that is, for every point ${\displaystyle x\in X,}$ the singleton set ${\displaystyle \{x\}}$ is a closed subset of ${\displaystyle X.}$
4. Every subset of ${\displaystyle X}$ is the intersection of all the open sets containing it.
5. Every finite set is closed.^[2]
6. Every cofinite set of ${\displaystyle X}$ is open.
7. For every ${\displaystyle x\in X,}$ the fixed ultrafilter at ${\displaystyle x}$ converges only to ${\displaystyle x.}$
8. For every subset ${\displaystyle S}$ of ${\displaystyle X}$ and every point ${\displaystyle x\in X,}$ ${\displaystyle x}$ is a limit point of ${\displaystyle S}$ if and only if every open neighbourhood of ${\displaystyle x}$ contains infinitely many points of ${\displaystyle S.}$
9. Each map from the Sierpiński space to ${\displaystyle X}$ is trivial.
10. The map from the Sierpiński space to the single point has the lifting property with respect to the map from ${\displaystyle X}$ to the single point.

If ${\displaystyle X}$ is a topological space then the following conditions are equivalent:^[3] (where ${\displaystyle \operatorname {cl} \{x\}}$ denotes the closure of ${\displaystyle \{x\}}$)

1. ${\displaystyle X}$ is an R₀ space.
2. Given any ${\displaystyle x\in X,}$ the closure of ${\displaystyle \{x\}}$ contains only the points that are topologically indistinguishable from ${\displaystyle x.}$
3. The Kolmogorov quotient of ${\displaystyle X}$ is T₁.
4. For any ${\displaystyle x,y\in X,}$ ${\displaystyle x}$ is in the closure of ${\displaystyle \{y\}}$ if and only if ${\displaystyle y}$ is in the closure of ${\displaystyle \{x\}.}$
5. The specialization preorder on ${\displaystyle X}$ is symmetric (and therefore an equivalence relation).
6. The sets ${\displaystyle \operatorname {cl} \{x\}}$ for ${\displaystyle x\in X}$ form a partition of ${\displaystyle X}$ (that is, any two such sets are either identical or disjoint).
7. If ${\displaystyle F}$ is a closed set and ${\displaystyle x}$ is a point not in ${\displaystyle F}$, then ${\displaystyle F\cap \operatorname {cl} \{x\}=\emptyset .}$
8. Every neighbourhood of a point ${\displaystyle x\in X}$ contains ${\displaystyle \operatorname {cl} \{x\}.}$
9. Every open set is a union of closed sets.
10. For every ${\displaystyle x\in X,}$ the fixed ultrafilter at ${\displaystyle x}$ converges only to the points that are topologically indistinguishable from ${\displaystyle x.}$

In any topological space we have, as properties of any two points, the following implications

separated ${\displaystyle \implies }$ topologically distinguishable ${\displaystyle \implies }$ 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. 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).

A space that is locally T₁, in the sense that each point has a T₁ neighbourhood (when given the subspace topology), is also T₁.^[4] Similarly, a space that is locally R₀ is also R₀. In contrast, the corresponding statement does not hold for T₂ spaces. For example, the line with two origins is not a Hausdorff space but is locally Hausdorff.

• Sierpiński 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 nonempty open sets of the cofinite topology are disjoint. Specifically, let ${\displaystyle X}$ be the set of integers, and define the open sets ${\displaystyle O_{A}}$ to be those subsets of ${\displaystyle X}$ that contain all but a finite subset ${\displaystyle A}$ of ${\displaystyle X.}$ Then given distinct integers ${\displaystyle x}$ and ${\displaystyle y}$:

• the open set ${\displaystyle O_{\{x\}}}$ contains ${\displaystyle y}$ but not ${\displaystyle x,}$ and the open set ${\displaystyle O_{\{y\}}}$ contains ${\displaystyle x}$ and not ${\displaystyle y}$;
• equivalently, every singleton set ${\displaystyle \{x\}}$ is the complement of the open set ${\displaystyle O_{\{x\}},}$ 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 intersection of any two open sets ${\displaystyle O_{A}}$ and ${\displaystyle O_{B}}$ is ${\displaystyle O_{A}\cap O_{B}=O_{A\cup B},}$ 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 ${\displaystyle X}$ be the set of integers again, and using the definition of ${\displaystyle O_{A}}$ from the previous example, define a subbase of open sets ${\displaystyle G_{x}}$ for any integer ${\displaystyle x}$ to be ${\displaystyle G_{x}=O_{\{x,x+1\}}}$ if ${\displaystyle x}$ is an even number, and ${\displaystyle G_{x}=O_{\{x-1,x\}}}$ if ${\displaystyle x}$ is odd. Then the basis of the topology are given by finite intersections of the subbasic sets: given a finite set ${\displaystyle A,}$the open sets of ${\displaystyle X}$ are

${\displaystyle U_{A}:=\bigcap _{x\in A}G_{x}.}$

The resulting space is not T₀ (and hence not T₁), because the points ${\displaystyle x}$ and ${\displaystyle x+1}$ (for ${\displaystyle x}$ 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) is T₁. To see this, note that the singleton containing a point with local coordinates ${\displaystyle \left(c_{1},\ldots ,c_{n}\right)}$ is the zero set of the polynomials ${\displaystyle x_{1}-c_{1},\ldots ,x_{n}-c_{n}.}$ 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 (that is, the prime spectrum of a ring) is T₀ but not, in general, T₁.^[5] To see this, note that the closure of a one-point set is the set of all prime ideals 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 ${\displaystyle A}$ is given as follows: the topological space is the set ${\displaystyle X}$ of all prime ideals of ${\displaystyle A.}$ The base of the topology is given by the open sets ${\displaystyle O_{a}}$ of prime ideals that do not contain ${\displaystyle a\in A.}$ It is straightforward to verify that this indeed forms the basis: so ${\displaystyle O_{a}\cap O_{b}=O_{ab}}$ and ${\displaystyle O_{0}=\varnothing }$ and ${\displaystyle O_{1}=X.}$ The closed sets of the Zariski topology are the sets of prime ideals that do contain ${\displaystyle a.}$ 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 space is T₁, since every point is a connected component and therefore closed.

The terms "T₁", "R₀", and their synonyms can also be applied to such variations of topological spaces as uniform spaces, Cauchy spaces, 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.

• Topological property – Mathematical property of a space

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