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T1 space : Symmetric 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.

Definitions

A topological space X is T₁ (also called accessible or Fréchet) 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 doesn't 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.

Each of the above conditions implies the other one; so whichever one is most convenient can be used.

X is R₀ (also called symmetric), 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 doesn't 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,...,c_n) is the zero set[?] of the polynomials x₁-c₁, ..., x_n-c_n. Thus, the point is closed. However, this example is well known as a space that isn't 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 O_A 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 isn't T₂, because the intersection of any two open sets O_A and O_B is O_A∪B, 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 O_A from the previous example, define a basis of open sets G_x for any integer x to be G_x = O_{{x, x+1}} if x is an even number, and G_x = O_{{x-1, x}} if x is odd. Then the open sets of X are, unions of the basis sets

U_A := ∪_{x in A} G_x.

The resulting space isn't 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 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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