# Diffeological vector spaces

## TL;DR

This paper explores the relationships between various natural conditions on diffeological vector spaces, establishing a hierarchy and providing examples to clarify which conditions imply others, with applications to homological algebra.

## Contribution

It characterizes the logical relationships among key properties of diffeological vector spaces and investigates their implications for homological algebra.

## Key findings

- Most conditions form a total order in the hierarchy
- Examples show which implications do not hold
- Results inform the homological algebra of diffeological vector spaces

## Abstract

We study the relationship between many natural conditions that one can put on a diffeological vector space: being fine or projective, having enough smooth (or smooth linear) functionals to separate points, having a diffeology determined by the smooth linear functionals, having fine finite-dimensional subspaces, and having a Hausdorff underlying topology. Our main result is that the majority of the conditions fit into a total order. We also give many examples in order to show which implications do not hold, and use our results to study the homological algebra of diffeological vector spaces.