# fully T4

A topological space^{} $X$ is said to be *fully* ${T}_{4}$ if every open cover of $X$
has star refinement.

A topological space is said to be *fully normal* if it is a ${T}_{1}$ space and
is fully ${T}_{4}$.

For example, every pseudometric space is fully ${T}_{4}$.

We have the following implications^{}:

Lindelöf ${T}_{3}\Rightarrow $ paracompact and ${T}_{3}\Rightarrow $ fully ${T}_{4}\Rightarrow {T}_{4}\Rightarrow $ uniformizable $\Rightarrow {T}_{3}$ ,

and

fully normal $\iff $ paracompact regular^{}.

Title | fully T4 |
---|---|

Canonical name | FullyT4 |

Date of creation | 2013-03-22 17:09:43 |

Last modified on | 2013-03-22 17:09:43 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 7 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 54D15 |

Defines | fully normal |