# Hausdorff Gaps and Limits

**By**

- R. Frankiewicz, Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
- P. Zbierski, Department of Mathematics, University of Warsaw, Warsaw, Poland

Gaps and limits are two phenomena occuring in the Boolean algebra *P*(&ohgr;)/fin. Both were discovered by F. Hausdorff in the mid 1930's. This book aims to show how they can be used in solving several kinds of mathematical problems and to convince the reader that they are of interest in themselves. The forcing technique, which is not commonly known, is used widely in the text. A short explanation of the forcing method is given in Chapter 11. Exercises, both easy and more difficult, are given throughout the book.

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### Book information

- Published: February 1994
- Imprint: NORTH-HOLLAND
- ISBN: 978-0-444-89490-8

### Table of Contents

Notation and terminology. **1. Boolean Algebras. **Introduction. Formulas. Atoms. Complete algebras. Homomorphism and filters. Ultrafilters. Extending a homomorphism. Chains and antichains. Problems. **2. Gaps and Limits. **Introduction. Dominance. Hausdorff Gaps. The Parovičenko theorem. Types of gaps and limits. Problems. **3. Stone Spaces. **The Stone Representation. Subalgebras and homomorphisms. Zero-sets. The Stone-Čech compactification. Spaces of uniform ultrafilters. Strongly zero-dimensional spaces. Extremally disconnected spaces. Problems. **4. F-Spaces. **Extending a function. Characterization of countable gaps. Construction of Parovičenko spaces. Closed sets in the space &ohgr;*. On the Parovičenko theorem. On P-sets in the space &ohgr;*. Character of points. Problems. **5. &pgr;-Base Matrix. ** Base tree. Stationary sets. &sfgr;-Points. Problems. **6. Inhomogeneity. **Kunen's points. A matrix of independent sets. Countable sets in F-spaces. Inhomogeneity of products of compact spaces. Problems. **7. Extending of Continuous Functions. **Weak Lindelhöf property. A long convergent sequence. Strongly discrete sets. Problems. **8. The Martin Axiom. **Continuous images. The space &bgr;[&ohgr;_{1}]. On the Parovičenko theorem. Gaps. Homomorphisms of C(X). Problems. **9. Partitions of Antichains. **Partition of algebras. Complete algebras. Partition algebras under MA. More on partition algebras. Problems. **10. Small P-Sets in &ohgr;*. **Proper forcing. On P-filters with the ccc. Problems. **11. Forcing. **Set theory and its models. Forcing. Complete embeddings. Cardinal numbers. Selected models. Iterated forcing. The Martin Axiom. Bibliography. Index