T1 space

Last updated
Separation axioms
in topological spaces
Kolmogorov classification
T0  (Kolmogorov)
T1  (Fréchet)
T2  (Hausdorff)
T2½ (Urysohn)
completely T2  (completely Hausdorff)
T3  (regular Hausdorff)
T3½ (Tychonoff)
T4  (normal Hausdorff)
T5  (completely normal
 Hausdorff)
T6  (perfectly normal
 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. [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.

Contents

Definitions

Let X be a topological space and let x and y be points in X. We say that x and y can be separated if each lies in a neighborhood that does not contain the other point.

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 has another meaning.)

Properties

If X is a topological space then the following conditions are equivalent:

  1. X is a T1 space.
  2. X is a T0 space and an R0 space.
  3. Points are closed in X; i.e. given any the singleton set is a closed set.
  4. Every subset of X is the intersection of all the open sets containing it.
  5. Every finite set is closed. [2]
  6. Every cofinite set of X is open.
  7. The fixed ultrafilter at x converges only to x.
  8. For every subset S of X and every point x is a limit point of S if and only if every open neighbourhood of x contains infinitely many points of S.

If X is a topological space then the following conditions are equivalent:

  1. X is an R0 space.
  2. Given any the closure of contains only the points that are topologically indistinguishable from x.
  3. For any two points x and y in the space, x is in the closure of if and only if y is in the closure of
  4. The specialization preorder on X is symmetric (and therefore an equivalence relation).
  5. The fixed ultrafilter at x converges only to the points that are topologically indistinguishable from x.
  6. Every open set is the union of closed sets.

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

separatedtopologically distinguishabledistinct

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's both R0 and T0.

Note that a finite T1 space is necessarily discrete (since every set is closed).

Examples

  • 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 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.

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.

See also

Citations

  1. Arkhangel'skii (1990). See section 2.6.
  2. Archangel'skii (1990) See proposition 13, section 2.6.
  3. Arkhangel'skii (1990). See example 21, section 2.6.

Bibliography

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