Accessible space
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Separation axioms in topological spaces |
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|---|---|
| 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) |
| History | |
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. 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 which does not contain the other point.
A T1 space is also called an accessible space or a Tikhonov space, or a space with Fréchet topology and a 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.)
Let X be a topological space. Then the following conditions are equivalent:
Let X be 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. Clearly, a space is T1 if and only if it's both R0 and T0.
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