A suggestion towards a finitist's realisation of topology

TL;DR

This paper explores a finitist approach to topology by characterizing key topological properties using a single morphism and Quillen lifting properties, aiming for a combinatorial framework.

Contribution

It introduces a finitistic, combinatorial perspective on topology, defining trivial fibrations and contractibility via a single finite topological space morphism.

Findings

Characterizes trivial Serre fibrations using Quillen lifting properties.

Identifies a single morphism that captures key topological properties.

Suggests a potential finitistic model structure for topological spaces.

Abstract

We observe that the notion of a trivial Serre fibration, a Serre fibration, and being contractible, for finite CW complexes, can be defined in terms of the Quillen lifting property with respect to a single map M-->/\ of finite topological spaces (preorders) of size 5 and 3. In particular, we observe that the double Quillen orthogonal { M-->/\ }^lr is precisely the class of trivial Serre fibrations if calculated in a certain category of nice topological spaces. This suggests a question whether there is a finitistic/combinatorial definition of a model structure on the category of topological spaces entirely in terms of the single morphism M-->/\, apparently related to the Michael continuous selection theory.