# separation axioms

The *separation axioms ^{}* are additional conditions which may be required to a topological space

^{}in order to ensure that some particular types of sets can be separated by open sets, thus avoiding certain pathological cases.

Axiom | Definition |
---|---|

${T}_{0}$ | given two distinct points, there is an open set containing exactly one of them; |

${T}_{1}$ (http://planetmath.org/T1Space) | given two distinct points, there is a neighborhood^{} of each of them which does not contain the other point; |

${T}_{2}$ (http://planetmath.org/T2Space) | given two distinct points, there are two disjoint open sets each of which contains one of the points; |

${T}_{2\u2064\frac{1}{2}}$ | given two distinct points, there are two open sets, each of which contains one of the points, whose closures^{} are disjoint; |

${T}_{3}$ (http://planetmath.org/T3Space) | given a closed set^{} $A$ and a point $x\notin A$, there are two disjoint open sets $U$ and $V$ such that $x\in U$ and $A\subset V$; |

${T}_{3\u2064\frac{1}{2}}$ | given a closed set $A$ and a point $x\notin A$, there is an Urysohn function for $A$ and $\{b\}$; |

${T}_{4}$ | given two disjoint closed sets $A$ and $B$, there are two disjoint open sets $U$ and $V$ such that $A\subset U$ and $B\subset V$; |

${T}_{5}$ | given two separated sets $A$ and $B$, there are two disjoint open sets $U$ and $V$ such that $A\subset U$ and $B\subset V$. |

If a topological space satisfies a ${T}_{i}$ axiom, it is called a ${T}_{i}$-space. The following table shows other common names for topological spaces with these or other additional separation properties.

Name | Separation properties |
---|---|

Kolmogorov space | ${T}_{0}$ |

Fréchet space | ${T}_{1}$ |

Hausdorff space | ${T}_{2}$ |

Completely Hausdorff space | ${T}_{2\u2064\frac{1}{2}}$ |

Regular space^{} |
${T}_{3}$ and ${T}_{0}$ |

Tychonoff^{} or completely regular space |
${T}_{3\u2064\frac{1}{2}}$ and ${T}_{0}$ |

Normal space | ${T}_{4}$ and ${T}_{1}$ |

Perfectly ${T}_{4}$ space | ${T}_{4}$ and every closed set is a ${G}_{\delta}$ (see here (http://planetmath.org/G_deltaSet)) |

Perfectly normal space | ${T}_{1}$ and perfectly ${T}_{4}$ |

Completely normal space | ${T}_{5}$ and ${T}_{1}$ |

The following implications^{} hold strictly:

$({T}_{2}\text{and}{T}_{3})$ | $\Rightarrow {T}_{2\u2064\frac{1}{2}}$ | ||

$({T}_{3}\text{and}{T}_{4})$ | $\Rightarrow {T}_{3\u2064\frac{1}{2}}$ | ||

${T}_{3\u2064\frac{1}{2}}$ | $\Rightarrow {T}_{3}$ | ||

${T}_{5}$ | $\Rightarrow {T}_{4}$ |

Completely normal | $\Rightarrow \text{normal}\Rightarrow \text{completely regular}$ | ||

$\Rightarrow \text{regular}\Rightarrow {T}_{2\u2064\frac{1}{2}}\Rightarrow {T}_{2}\Rightarrow {T}_{1}\Rightarrow {T}_{0}$ |

Remark. Some authors define ${T}_{3}$ spaces in the way we defined regular spaces, and ${T}_{4}$ spaces in the way we defined normal spaces (and vice-versa); there is no consensus on this issue.

Bibliography: *Counterexamples in Topology*, L. A. Steen, J. A. Seebach Jr., Dover Publications Inc. (New York)

Title | separation axioms |

Canonical name | SeparationAxioms |

Date of creation | 2013-03-22 13:28:47 |

Last modified on | 2013-03-22 13:28:47 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 26 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 54D10 |

Classification | msc 54D15 |

Synonym | separation properties |

Related topic | NormalTopologicalSpace |

Related topic | HausdorffSpaceNotCompletelyHausdorff |

Related topic | SierpinskiSpace |

Related topic | MetricSpacesAreHausdorff |

Related topic | ZeroDimensional |

Related topic | T2Space |

Related topic | RegularSpace |

Related topic | T4Space |

Defines | Hausdorff |

Defines | completely Hausdorff |

Defines | normal |

Defines | completely normal |

Defines | regular^{} |

Defines | Tychonoff |

Defines | completely regular |

Defines | perfectly normal |

Defines | Tychonov |

Defines | perfectly ${T}_{4}$ |