| Separation axioms |
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.
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.)
If X is a topological space then the following conditions are equivalent:
If X is a topological space then the following conditions are equivalent:
In any topological space we have, as properties of any two points, the following implications
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).
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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