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# How are normal and T4 spaces defined in books?

A recent discussion on PlanetMath has led me to consider how various sources define normal and T4 spaces. I limited myself to books, mostly textbooks. No articles were consulted. As will be seen from the table below, there is no agreement on the question of how to define it. I am not giving precise references at this time, and may choose to never do that. I think the abbreviated form may be sufficient for those that seek to check what I have done. If you want to add something to the table, file a correction. S refers to the condition that closed sets can be separated by open sets. The condition S is due to Tietze, according to Alexandroff and Hopf. Of course, T1 + S is the same as T2 +S.

Source | Normal | T4 | Page | Year |
---|---|---|---|---|

Alexandroff and Hopf | T1+S | T1+S | 68 | 1935 |

Wilder | S | ? | 49 | 1949 |

Kelley | S | T1+S | 112 | 1955 |

Hocking and Young | T1+S | T1+S | 41 | 1961 |

Pervin | S | T1+S | 88 | 1964 |

Gaal | T1+S | S | 87 | 1964 |

Lipschutz | S | T1+S | 141 | 1965 |

Husain | T1+S | S | 7 | 1966 |

Dugundji | T2+S | T2+S | 144 | 1966 |

Gemignani | T1+S | S | 102 | 1967 |

Willard | S | T1+S | 99 | 1970 |

Steen and Seebach | T1+S | S | 12 | 1970 |

Maunder | S | - | 15 | 1970 |

Munkres | T1+S | - | 195 | 1975 |

Morris | S | T2+S | 121 | 1988 |

Repovš | T1+S | S | 6 | 1998 |

Stroppel | T2+S | S | 6 | 2006 |

Related:

UrysohnsLemma , T4Space

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54D15*no label found*

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