## Abstract

It is consistent with the axioms of set theory that for every metric space X which is isometric to some separable Banach space or to Urysohn's universal separable metric space U the following holds: (*)X There exists a nowhere meager subspace of X of cardinality N_{1} and any two nowhere meager subsets of X of cardinality N_{1} are almost isometric to each other. As a corollary, it is consistent that the Continuum Hypothesis fails and the following hold: (1) There exists an almost-isometry ultrahomogeneous and universal element in the class of separable metric spaces of size N_{1}. (2) For every separable Banach space X there exists an almost-isometry conditionally ultrahomogeneous and universal element in the class of subspaces of X of size N_{1}. (3) For every finite dimensional Banach space X, there is a unioue universal element up to almost-isometry in the class of subspaces of X of size N_{1}

Original language | English |
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Pages (from-to) | 215-226 |

Number of pages | 12 |

Journal | Mathematical Research Letters |

Volume | 14 |

Issue number | 2-3 |

DOIs | |

State | Published - 1 Jan 2007 |

## Keywords

- Almost-isometric embedding
- Almost-isometry
- Metric space
- Oracle forcing
- Universality
- Urysohn's space

## ASJC Scopus subject areas

- Mathematics (all)