Finite combinatorics implicit in the basic definitions of topology

TL;DR

This paper reveals how finite combinatorics underpins basic topological concepts, using a concise, homotopy-theoretic notation based on category-theoretic lifting properties, making definitions more compact and accessible.

Contribution

It introduces a novel combinatorial notation that simplifies the definitions of key topological properties using finite preorders and category-theoretic concepts.

Findings

Most topological properties can be characterized by finite spaces of size at most 5.

The lifting property serves as a powerful tool for defining fundamental concepts.

The notation reduces complex definitions to 2 or 4 bytes.

Abstract

We explain how to see finite combinatorics of preorders implicit in the {text} of basic topological definitions or arguments in (Bourbaki, General topology, Ch.I), and define a concise combinatorial notation such that complete definitions of connectedness, compactness, contractibility, having a generic point, subspace, closed subspace, fit into $2$ or $4$ bytes. This notation is homotopy theoretic in nature, and is based on the following observation: A number of basic properties of continuous maps and topological spaces are defined using a single category-theoretic operation, taking left or right orthogonal complement with respect to the Quillen lifting property, repeatedly applied to a simple example illustrating the definition or its failure. Moreover, for most of these definitions this example can be chosen to be a map of finite topological spaces (=preorders) of size at most $5$. This includes the properties of a space being connected, compact, contractible, discrete, having a generic point, and a map having dense image, being the inclusion of an (open or closed) subspace, or of a component into a disjoint union, and others. Our reformulations illustrate the generative power of the lifting property as a means of defining basic mathematical properties starting from their simplest or typical example. The exposition is accessible to a student.