Algebra and Metrizability of Uniform Spaces

Abstract

This thesis is an exploration into the concept of a uniform space using algebraic and category-theoretic techniques usually applied to topological spaces. The hope is that we can help future research to know more about the nature of uniform continuity from this perspective.

The point-free perspective of the lattice of open sets of a topological space is applied to give another lattice, of entourages of the uniform space. A binary operation of entourage addition or composition is fundamental to the definition of a uniform space, and equipping the lattice of entourages with this binary operation yields an algebraic structure known by several names, one of which being a Heyting Algebra, but which we denote by the name “uniform frame” for consistency in terminology.

In the first two chapters, we establish some definitions and concepts that will be of use throughout the paper. In the third chapter, we prove a precise condition for the metrizability of a uniform space, which is equivalent to a known condition, but given in a more order-theoretic and category-theoretic language. In the fourth chapter, the relation between a uniform space and its induced topology is defined and explored. We stack the uniform frame of covers on top of the frame of open sets and observe the presence of a canonical projection morphism from the former to the latter, and this morphism is denoted as a uniform setting. We conjecture that every uniform space can be reconstructed from its uniform setting.