## Set Theory and Metric SpacesThis is a book that could profitably be read by many graduate students or by seniors in strong major programs ... has a number of good features. There are many informal comments scattered between the formal development of theorems and these are done in a light and pleasant style. ... There is a complete proof of the equivalence of the axiom of choice, Zorn's Lemma, and well-ordering, as well as a discussion of the use of these concepts. There is also an interesting discussion of the continuum problem ... The presentation of metric spaces before topological spaces ... should be welcomed by most students, since metric spaces are much closer to the ideas of Euclidean spaces with which they are already familiar. --Canadian Mathematical Bulletin Kaplansky has a well-deserved reputation for his expository talents. The selection of topics is excellent. --Lance Small, UC San Diego This book is based on notes from a course on set theory and metric spaces taught by Edwin Spanier, and also incorporates with his permission numerous exercises from those notes. The volume includes an Appendix that helps bridge the gap between metric and topological spaces, a Selected Bibliography, and an Index. |

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### Contents

CARDINAL NUMBERS | 21 |

WELLORDERING THE AXIOM OF CHOICE | 49 |

BASIC PROPERTIES OF METRIC SPACES | 67 |

COMPLETENESS SEPARABILITY AND COMPACTNESS | 84 |

ADDITIONAL TOPICS | 106 |

Examples of Metric Spaces | 117 |

The Transition to Topological Spaces | 127 |

133 | |

### Common terms and phrases

abelian group assume axiom of choice called Cantor cardinal number Cartesian product Cauchy sequence chain closed ball closed sets closed subset closure compact metric space complement complete continuous function continuum hypothesis contradiction countable set countably infinite defined DEFINITION dense subset descending sequence diam disjoint distance function example exists a one-to-one finite number given greatest lower bound Hence Hint ideal implies infinite cycle infinite set infinite subset intersection isometry lattice least upper bound limit point maximal element neighborhood nonempty notation one-to-one correspondence one-to-one map open ball open covering open sets order-isomorphic ordinal partially ordered set pick positive integers power set proof of Theorem Prove rational numbers real line Remark satisfying Section 4.l segment set theory set with cardinal set-theoretic smallest element statements Suppose Theorem l0 topological spaces totally unordered transfinite triangle inequality uncountable uniformly continuous union unique well-ordered set write Zorn's lemma