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A (Hausdorf) hereditarily normal (not perfectly normal) space  X  is constructed, which has the following properties: (a) there exists a proper open subspace of  X  which is homeomorphic to the whole  X  (i.e., the space  X  is topologically infinite); (b) the space is homeomorphic to none of its properF  σ{F_{\sigma}}   -subspaces (i.e., the space  X  is topologically finite relative to the class of all its properF  σ{F_{\sigma}}   -subspaces).

georgian mathematical journal

An example of a hereditarily normal topologically finite space, which is topologically infinite relative to the class of all its proper  F  σ -subspaces