# New results regarding the lattice of uniform topologies on $C(X)$

## TL;DR

This paper investigates the structural, categorical, and cardinal properties of the lattice of uniform topologies on the space of continuous functions over a Tychonoff space, extending previous work in the area.

## Contribution

It advances understanding of the lattice $al U_X$ by exploring its structural, categorical, and cardinal characteristics and their relationships to properties of the underlying space.

## Key findings

- Analyzed the structural properties of $al U_X$.
- Established connections between lattice cardinal characteristics and space cardinal functions.
- Provided new insights into the categorical aspects of the lattice of uniform topologies.

## Abstract

For a Tychonoff space $X$, the lattice $\mathscr U_X$ was introduced in L.A. P\'erez-Morales, G. Delgadillo-Pi\~n\'on, and R. Pichardo-Mendoza, "The lattice of uniform topologies on $C(X)$", Questions and Answers in General Topology, 39 (2021), 65-71. In the present paper we continue the study of $\mathscr U_X$. To be specific, the present paper deals, in its first half, with structural and categorical properties of $\mathscr U_X$, while in its second part focuses on cardinal characteristics of the lattice and how these relate to some cardinal functions of the space $X$.