T1 space

In topology and related branches of mathematics, T1 spaces and R0 spaces are particularly nice kinds of topological spaces. The T1 and R0 properties are examples of separation axioms.

A T1 space is also called an accessible space or a Fréchet space and a R0 space is also called an symmetric space.

Table of contents
1 Definitions
2 Examples
3 Generalisations to other kinds of spaces

Definitions

A topological space X is T1 if and only if either of the following equivalent conditions is satisfied:

  • Given any two distinct points x and y in X, each lies in an open set which does not contain the other. In other words, the singleton sets {x} and {y} are separated unless x = y.
  • Given any x in X, {x} is a closed set. In other words, the fixed ultrafilter at x converges only to x.
  • For every point x in X and every subset S of X, x is a limit point of S if and only if every open neighbourhood of x contains infinitely many points of S.
    • Proof: Suppose singletons are closed in X. Let S be a subset of X and x a limit point of S. Suppose there is an open neighbourhood U of x that contains only finitely many points of S. Then U \\ (S \\ {x}) is an open neighbourhood of x that does not contain any points of S other than x. (Here is where we use the fact that singletons are closed.) This contradicts the fact that x is a limit point of S. Thus, every open neighbourhood of x contains infinitely many points of S. Conversely, suppose there is a point x in X such that the singleton {x} is not closed. Then there is a point yx in the closure of {x}. We claim that any open neighbourhood U of y contains x. For suppose not; then the complement of U in X would be a closed set containing x, and the closure of {x} would be contained in the complement of U. Since y is in the closure of {x}, this would force y not to be in U, contradicting the fact that U is a neighbourhood of y. We have shown that y is a limit point of S = {x}. But it is clear that X is a neighbourhood of y that does not contain infinitely many points of S. This completes the proof.

X is R0 if and only if either of the following conditions is satisfied:
  • Given any two topologically distinguishable points x and y in X, each lies in an open set which does not contain the other. In other words, {x} and {y} are separated unless x and y are topologically indistinguishable.
  • Given any x in X, the closure of {x} owns only the points that x is topologically indistinguishable from. In other words, the fixed ultrafilter at x converges only to the points that x is topologically indistinguishable from.
As before, the above conditions are equivalent.

A space is T1 if and only if it's both R0 and T0 (which says that topologically indistinguishable points must be equal). Conversely, a space is R0 if and only if its Kolmogorov quotient (which identifies topologially indistinguishable points) is T1.

Do not confuse the term "Fréchet topology", which is equivalent to "T1 topology", with the term "Fréchet space" which refers to an entirely different notion from functional analysis.

Examples

The Zariski topology on an algebraic variety is T1. To see this, note that a point with local coordinates (c1,...,cn) is the zero set of the polynomials x1-c1, ..., xn-cn. Thus, the point is closed. However, this example is well known as a space that is not Hausdorff (T2).

For a more concrete example, let's look at the cofinite topology on an infinite set. Specifically, let X be the set of integers, and define the open sets OA to be those subsets of X which contain all but a finite subset A of X. Then given distinct integers x and y:

  • the open set O{x} contains y but not x, and the open set O{y} contains x and not y;
  • equivalently, every singleton set {x} is the complement of the open set 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 OA and OB is OAB, which is never empty. Alternatively, the set of even integers is compact but not closed, which would be impossible in a Hausdorff space.

We can modify this example slightly to get an R0 space that is neither T1 nor R1. Let X be the set of integers again, and using the definition of OA from the previous example, define a basis of open sets Gx for any integer x to be Gx = O{x, x+1} if x is an even number, and Gx = O{x-1, x} if x is odd. Then the open sets of X are, unionss of the basis sets

UA := x in A Gx.
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.

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 netss) 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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