in topological spaces
|completely T2||(completely Hausdorff)|
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. 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.
- 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 is a topological space then the following conditions are equivalent:
- is a T1 space.
- is a T0 space and an R0 space.
- Points are closed in ; that is, for every point the singleton set is a closed subset of
- Every subset of is the intersection of all the open sets containing it.
- Every finite set is closed.
- Every cofinite set of is open.
- For every the fixed ultrafilter at converges only to
- For every subset of and every point is a limit point of if and only if every open neighbourhood of contains infinitely many points of
- Each map from the Sierpiński space to is trivial.
- The map from the Sierpiński space to the single point has the lifting property with respect to the map from to the single point.
If is a topological space then the following conditions are equivalent: (where denotes the closure of )
- is an R0 space.
- Given any the closure of contains only the points that are topologically indistinguishable from
- The Kolmogorov quotient of is T1.
- For any is in the closure of if and only if is in the closure of
- The specialization preorder on is symmetric (and therefore an equivalence relation).
- The sets for form a partition of (that is, any two such sets are either identical or disjoint).
- If is a closed set and is a point not in , then
- Every neighbourhood of a point contains
- Every open set is a union of closed sets.
- For every the fixed ultrafilter at converges only to the points that are topologically indistinguishable from
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. 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. 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 be the set of integers, and define the open sets to be those subsets of that contain all but a finite subset of Then given distinct integers and :
- the open set contains but not and the open set contains and not ;
- equivalently, every singleton set is the complement of the open set 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 and is 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 be the set of integers again, and using the definition of from the previous example, define a subbase of open sets for any integer to be if is an even number, and if is odd. Then the basis of the topology are given by finite intersections of the subbasic sets: given a finite set the open sets of are
- The resulting space is not T0 (and hence not T1), because the points and (for 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 is the zero set of the polynomials 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. 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 is given as follows: the topological space is the set of all prime ideals of The base of the topology is given by the open sets of prime ideals that do not contain It is straightforward to verify that this indeed forms the basis: so and and The closed sets of the Zariski topology are the sets of prime ideals that do contain 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.
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 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.
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