Redirected from Symmetric space
A topological space X is T_{1} (also called accessible or Fréchet) if and only if either of the following equivalent conditions is satisfied:
X is R_{0} (also called symmetric), if and only if either of the following conditions is satisfied:
A space is T_{1} if and only if it's both R_{0} and T_{0} (which says that topologically indistinguishable points must be equal). Conversely, a space is R_{0} if and only if its Kolmogorov quotient (which identifies topologially indistinguishable points) is T_{1}.
Do not confuse the term "Fréchet topology", which is equivalent to "T_{1} topology", with the term "Fréchet space" which refers to an entirely different notion from functional analysis.
The Zariski topology on an algebraic variety is T_{1}. To see this, note that a point with local coordinates[?] (c_{1},...,c_{n}) is the zero set[?] of the polynomials x_{1}c_{1}, ..., x_{n}c_{n}. Thus, the point is closed. However, this example is well known as a space that is not Hausdorff (T_{2}).
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:
We can modify this example slightly to get an R_{0} space that is neither T_{1} nor R_{1}. 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_{{x1, x}} if x is odd. Then the open sets of X are, unions of the basis sets
Generalisations to other kinds of spaces
The terms "T_{1}", "R_{0}", 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 T_{1} spaces) or unique up to topological indistinguishability (for R_{0} spaces).
As it turns out, uniform spaces, and more generally Cauchy spaces, are always R_{0}, so the T_{1} condition in these cases reduces to the T_{0} condition. But R_{0} alone can be an interesting condition on other sorts of convergence spaces, such as pretopological spaces[?].
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