# Locally small spaces with an application

## TL;DR

This paper introduces a simplified framework for locally small spaces and applies it to reconstruct the theory of locally definable spaces over topological structures.

## Contribution

It develops a new simple language for locally small spaces and uses it to re-establish the theory of locally definable spaces.

## Key findings

- Simplified the theory of locally small spaces.
- Rebuilt the theory of locally definable spaces using the new language.
- Provided a unified approach to topological structures.

## Abstract

We develop the theory of locally small spaces in a new simple language and apply this simplification to re-build the theory of locally definable spaces over structures with topologies.