Jean E. Rubin

Equivalents of the axiom of choice, II

Set Forms. The Well-Ordering Theorem. The Axiom of Choice. The Law of the Trichotomy. Maximal Principles. Forms Equivalent to the Axiom of Choice Under the Axioms of Extensionality and Foundation. Algebraic Forms. Cardinal Number Forms. Forms from Topology, Analysis and Logic. Class Forms. The Well-Ordering Theorem. The Axiom of Choice. Maximal Principles. List of the Set Forms. List of the Class Forms. List of Forms Related to the Axiom of Choice. Bibliography. Index.

Compactness in Countable Tychonoff Products and Choice

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.

Disjoint Unions of Topological Spaces and Choice

We nd properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice. Introduction and Terminology This is a continuation of the study of the roll the Axiom of Choice plays in general topology. See also (vd), (gt), (wgt), and (hkrr). Our primary concern will be the use of the axiom of choice in proving properties of disjoint unions of topological spaces (See Denition 1, part 11.) For example, in set theory with choice the disjoint union of metrizable topological spaces is a metrizable topological space. The usual proof of this fact begins with the choice of metrics for the component spaces. We will show that the use of some form of choice cannot be avoided in this proof and in fact without choice the disjoint union of metrizable spaces may not even be metacompact. In section 1 we show that many assertions about disjoint unions of topological spaces are equivalent to the axiom of multiple choice. Models of set theory and corresponding independence results are described in section 2. In section 3, we study the roll the Axiom of Choice plays in the properties of disjoint unions of collectionwise Hausdor and collectionwise normal spaces. We begin with the denitions of the symbols and terms we will be using. Denition 1. 1. A familyK of subsets of a topological space (X;T )i sl.f .( locally nite) i each point of X has a neighborhood meeting a nite number of elements ofK. 2. X is paracompact i X is T2 and every open coverU of X has a l.f.o.r. (locally nite open renement)V.T hat is,V is a locally nite open cover ofX and every member ofV is included in a member ofU. 3. A familyK of subsets of X is p.f. (point nite) i each element of X belongs to only nitely many members ofK. 4. X is metacompact i each open coverU of X has an o.p.f.r. (open point nite renement). 5. An open coverU =fUi : i2 kg of X is shrinkable ithere exists an open cover V =fVi : i2 kg of non-empty sets such that V i Ui for all i2 k. V is also called a shrinking ofU. 6. X is a PFCS space i every p.f. open cover of X is shrinkable.

The Boolean Prime Ideal Theorem Plus Countable Choice Do Not Imply Dependent Choice

Two Fraenkel-Mostowski models are constructed in which the Boolean Prime Ideal Theorem is true. In both models, AC for countable sets is true, but AC for sets of cardinality 2 and the 2m = m principle are both false. The Principle of Dependent Choices is true in the first model, but false in the second. Mathematics Subject Classification: 03E25, 03E35, 04A25.

Non-constructive Properties of the Real Numbers

We study the relationship between various properties of the real numbers and weak choice principles.

Paracompactness of Metric Spaces and the Axiom of Multiple Choice

The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice.

NON-CONSTRUCTIVE PROPERTIES OF CARDINAL NUMBERS

Let Σ be some standard set theory (Eg. Zermelo Fraenkel or Von Neumann-Bernays-Godel) which does not contain the axiom of choice. Using Σ as the underlying set theory, we shall study operations on infinite cardinals, closely related to exponentiation, and compare the results with known results about exponentiation.

Versions of Normality and Some Weak Forms of the Axiom of Choice

We investigate the set theoretical strength of some properties of normality, in- cluding Urysohns Lemma, Tietze-Urysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of F subsets of normal spaces.

Kinna-Wagner Selection Principles, Axioms of Choice and Multiple Choice

We study the relationships between weakened forms of the Kinna-Wagner Selection Principle (KW), the Axiom of Choice (AC), and the Axiom of Multiple Choice (MC).

Metric spaces and the axiom of choice

We study conditions for a topological space to be metrizable, properties of metriz- able spaces, and the role the axiom of choice plays in these matters.