Intopology and related branches ofmathematics, aT₁ space is atopological space in which, for every pair of distinct points, each has aneighborhood not containing the other point.^[1] AnR₀ space is one in which this holds for every pair oftopologically distinguishable points. The properties T₁ and R₀ are examples ofseparation axioms.

LetX be atopological space and letx andy be points inX. We say thatx andy areseparated if each lies in aneighbourhood that does not contain the other point.

• X is called aT₁ space if any two distinct points inX are separated.
• X is called anR₀ space if any twotopologically distinguishable points inX are separated.

A T₁ space is also called anaccessible space or a space withFréchet topology and an R₀ space is also called asymmetric space. (The termFréchet space also has anentirely different meaning infunctional analysis. For this reason, the termT₁ space is preferred. There is also a notion of aFréchet–Urysohn space as a type ofsequential space. The termsymmetric space also hasanother meaning.)

A topological space is a T₁ space if and only if it is both an R₀ space and aKolmogorov (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 itsKolmogorov 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 aT₀ 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 aclosed subset of${\displaystyle X.}$
4. Every subset of${\displaystyle X}$ is the intersection of all the open sets containing it.
5. Everyfinite set is closed.^[2]
6. Everycofinite set of${\displaystyle X}$ is open.
7. For every${\displaystyle x\in X,}$ thefixed 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 alimit point of${\displaystyle S}$ if and only if every openneighbourhood of${\displaystyle x}$ contains infinitely many points of${\displaystyle S.}$
9. Each map from theSierpiński space to${\displaystyle X}$ is trivial.
10. The map from theSierpiński space to the single point has thelifting 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${\displaystyle \mathrm{cl}\{x\}}$ denotes the closure of${\displaystyle \{x\}}$)

1. ${\displaystyle X}$ is an R₀ space.
2. Given any${\displaystyle x\in X,}$ theclosure of${\displaystyle \{x\}}$ contains only the points that are topologically indistinguishable from${\displaystyle x.}$
3. TheKolmogorov 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. Thespecialization preorder on${\displaystyle X}$ issymmetric (and therefore anequivalence relation).
6. The sets${\displaystyle \mathrm{cl}\{x\}}$ for${\displaystyle x\in X}$ form apartition 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${\displaystyle F\cap \mathrm{cl}\{x\}=\mathrm{\varnothing }.}$
8. Everyneighbourhood of a point${\displaystyle x\in X}$ contains${\displaystyle \mathrm{cl}\{x\}.}$
9. Everyopen set is a union ofclosed 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

${\displaystyle \text{separated}⟹\text{topologically distinguishable}⟹\text{distinct.}}$

If the first arrow can be reversed the space is R₀. If the second arrow can be reversed the space isT₀. 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 necessarilydiscrete (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, theline with two origins is not aHausdorff 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.
• Theoverlapping interval topology is a simple example of a topology that is T₀ but is not T₁.
• Everyweakly Hausdorff space is T₁ but the converse is not true in general.
• Thecofinite topology on aninfinite set is a simple example of a topology that is T₁ but is notHausdorff (T₂). This follows since no two nonempty open sets of the cofinite topology are disjoint. Specifically, let${\displaystyle X}$ be the set ofintegers, and define theopen sets${\displaystyle O_A}$ to be those subsets of${\displaystyle X}$ that contain all but afinite 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 theintersection 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 iscompact but notclosed, which would be impossible in a Hausdorff space.

• The above example can be modified slightly to create thedouble-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 asubbase of open sets${\displaystyle G_x}$ for any integer${\displaystyle x}$ to be${\displaystyle G_x=O_{\{x,x+1\}}}$ if${\displaystyle x}$ is aneven number, and${\displaystyle G_x=O_{\{x-1,x\}}}$ if${\displaystyle x}$ is odd. Then thebasis of the topology are given by finiteintersections 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.

• TheZariski topology on analgebraic variety (over analgebraically closed field) is T₁. To see this, note that the singleton containing a point withlocal coordinates${\displaystyle \left(c_1,\dots ,c_n\right)}$ is thezero set of thepolynomials${\displaystyle x_1-c_1,\dots ,x_n-c_n.}$ Thus, the point is closed. However, this example is well known as a space that is notHausdorff (T₂). The Zariski topology is essentially an example of a cofinite topology.
• The Zariski topology on acommutative ring (that is, the primespectrum 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 allprime ideals that contain the point (and thus the topology is T₀). However, this closure is amaximal 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 allprime ideals of${\displaystyle A.}$ Thebase of the topology is given by the open sets${\displaystyle O_a}$ of prime ideals that donot 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 thatdo 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.
• Everytotally disconnected space is T₁, since every point is aconnected component and therefore closed.

The terms "T₁", "R₀", and their synonyms can also be applied to such variations of topological spaces asuniform spaces,Cauchy spaces, andconvergence spaces.The characteristic that unites the concept in all of these examples is that limits of fixed ultrafilters (or constantnets) 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 aspretopological spaces.

• Topological property – Mathematical property of a space

• A.V. Arkhangel'skii, L.S. Pontryagin (Eds.)General Topology I (1990) Springer-VerlagISBN 3-540-18178-4.
• Folland, Gerald (1999).Real analysis: modern techniques and their applications (2nd ed.). John Wiley & Sons, Inc. p. 116.ISBN 0-471-31716-0.
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• Properties of topological spaces
• Separation axioms

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