A definition of finiteness is a set-theoretical property of a set that, if the Axiom of Choice (A... more A definition of finiteness is a set-theoretical property of a set that, if the Axiom of Choice (AC) is assumed, is equivalent to stating that the set is finite; several such definitions have been studied over the years. In this article we introduce a framework for generating definitions of finiteness in a systematical way: basic definitions are obtained from properties of certain classes of binary relations, and further definitions are obtained from the basic ones by closing them under subsets or under quotients.
We study the relationships between definitions of compactness in topological spaces and the roll ... more We study the relationships between definitions of compactness in topological spaces and the roll the axiom of choice plays in these relationships.
We study conditions for a topological space to be metrizable, properties of metrizable spaces, an... more We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters.
This is a continuation of [dhhkr]. We study the Tychonoff Compactness Theorem for various definit... more This is a continuation of [dhhkr]. We study the Tychonoff Compactness Theorem for various definitions of compact and for various types of spaces, (first and second countable spaces, Hausdorff spaces, and subspaces of R κ ). We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.
We study statements about countable and well-ordered unions and their relation to each other and ... more We study statements about countable and well-ordered unions and their relation to each other and to countable and well-ordered forms of the axiom of choice. Using WO as an abbreviation for "well-orderable", here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the axiom of choice for WO families of WO sets does not imply that the countable union of countable sets is WO.
A definition of finiteness is a set-theoretical property of a set that, if the Axiom of Choice (A... more A definition of finiteness is a set-theoretical property of a set that, if the Axiom of Choice (AC) is assumed, is equivalent to stating that the set is finite; several such definitions have been studied over the years. In this article we introduce a framework for generating definitions of finiteness in a systematical way: basic definitions are obtained from properties of certain classes of binary relations, and further definitions are obtained from the basic ones by closing them under subsets or under quotients.
We study the relationships between definitions of compactness in topological spaces and the roll ... more We study the relationships between definitions of compactness in topological spaces and the roll the axiom of choice plays in these relationships.
We study conditions for a topological space to be metrizable, properties of metrizable spaces, an... more We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters.
This is a continuation of [dhhkr]. We study the Tychonoff Compactness Theorem for various definit... more This is a continuation of [dhhkr]. We study the Tychonoff Compactness Theorem for various definitions of compact and for various types of spaces, (first and second countable spaces, Hausdorff spaces, and subspaces of R κ ). We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.
We study statements about countable and well-ordered unions and their relation to each other and ... more We study statements about countable and well-ordered unions and their relation to each other and to countable and well-ordered forms of the axiom of choice. Using WO as an abbreviation for "well-orderable", here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the axiom of choice for WO families of WO sets does not imply that the countable union of countable sets is WO.
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