# Tychonoff space

From Topospaces

## Contents

## Definition

A topological space is termed a **Tychonoff space** if it satisfies the following equivalent conditions:

No. | Shorthand | A topological space is termed Tychonoff if ... | A topological space is termed Tychonoff if ... |
---|---|---|---|

1 | T1 and completely regular | it is both a T1 space and a completely regular space | points are closed in , and given any point and closed subset such that , there exists a continuous map such that and for all . |

2 | Hausdorff and completely regular | it is both a Hausdorff space and a completely regular space | it is Hausdorff, and given any point and closed subset such that , there exists a continuous map such that and for all . |

3 | has a compactification | there is a compact Hausdorff space having a dense subspace (with the subspace topology) homeomorphic to it. (note: T1 assumption redundant in this case) | there is a compact Hausdorff space and a dense subspace of such that is homeomorphic to . |

4 | contained in compact Hausdorff | it is homeomorphic to a subspace (not necessarily dense) of a compact Hausdorff space (note: T1 assumption redundant in this case). | there is a compact Hausdorff space and a subspace of such that is homeomoephic to . |

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

*In the T family (properties of topological spaces related to separation axioms), this is called:* T3.5

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

normal Hausdorff space | Hausdorff and a normal space: disjoint closed subsets can be separated via disjoint open subsets | |FULL LIST, MORE INFO | ||

compact Hausdorff space | Hausdorff and a compact space: every open cover has a finite subcover | via compact Hausdorff implies normal | Paracompact Hausdorff space|FULL LIST, MORE INFO | |

underlying space of T0 topological group | |FULL LIST, MORE INFO | |||

metrizable space | underlying topological space of a metric space | Paracompact Hausdorff space|FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

completely regular space | |FULL LIST, MORE INFO | |||

regular Hausdorff space | |FULL LIST, MORE INFO | |||

regular space | Regular Hausdorff space|FULL LIST, MORE INFO | |||

Hausdorff space | Functionally Hausdorff space, Regular Hausdorff space, Urysohn space|FULL LIST, MORE INFO |