# classification of topological properties according to behaviour under mapping

Topological properties may be classified by their behaviour with respect to mappings. The basis of such a classification is the following question: Given two topological spaces^{} $X$ and $Y$ and a continuous map $f:X\to Y$, can one infer that one of the spaces has a certain topological property from the fact that the other space has this property?

A trivial case of this question may be disposed of. If $f$ is a homeomorphism^{}, then the spaces $X$ and $Y$ cannot be distinguished using only the techniques of topology, and hence both spaces will have exactly the same topological properties.

To obtain a non-trivial classification, we must consider more general maps. Since every map may be expressed as the composition^{} of an inclusion and a surjection, it is natural to consider the cases where $f$ is an inclusion and where it is a surjection.

In the case of an inclusion, we can define the following classifications:

A property of a topological space is called hereditary if it is the case that whenever a space has that property, every subspace^{} of that space also has the same property.

A property of a topological space is called weakly hereditary if it is the case that whenever a space has that property, every *closed* subspace of that space also has the same property.

In the case of a surjection, we can define the following classifications:

A property of a topological space is called continuous if it is the case that, whenever a space has this property, the images of this space under all continuous mapping also have the same property.

A property of a topological space is called open if it is the case that, whenever a space has this property, the images of this space under all open continuous mappings also have the same property.

A property of a topological space is called closed invariant if it is the case that, whenever a space has this property, the images of this space under all closed continuous mapping also have the same property.

Title | classification of topological properties according to behaviour under mapping |

Canonical name | ClassificationOfTopologicalPropertiesAccordingToBehaviourUnderMapping |

Date of creation | 2013-03-22 14:38:04 |

Last modified on | 2013-03-22 14:38:04 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 15 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 54C05 |

Defines | hereditary |

Defines | hereditarily |

Defines | weakly hereditary |

Defines | continuous |

Defines | open |

Defines | closed invariant |