Beta Spaces: New Generalizations of Typically-Metric Properties

Details Beta Spaces: New Generalizations of Typically-Metric Properties Beta Spaces: New Generalizations of Typically-Metric Properties

2010-11-17

arxiv.org/abs/1011.4027v1

Mathematics

General Topology

PREPRINT, submitted to arXiv for NSF review

Scientific paper

It is well-known that point-set topology (without additional structure) lacks the capacity to generalize the analytic concepts of completeness, boundedness, and other typically-metric properties. The ability of metric spaces to capture this information is tied to the fact that the topology is generated by open balls whose radii can be compared. In this paper, we construct spaces that generalize this property, called $\beta$-spaces, and show that they provide a framework for natural definitions of the above concepts. We show that $\beta$-spaces are strictly more general than uniform spaces, a common generalization of metric spaces. We then conclude by proving generalizations of several typically-metric theorems, culminating in a broader statement of the Contraction Mapping Theorem.

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