I often come across materials discussing convergence spaces and their relevance in various contexts. It's commonly mentioned that the existence of a natural convergence on the space of continuous functions (turning them into exponential objects) makes the category of these spaces a suitable environment for studying homotopy. However, I've found only a limited number of resources that actually delve into this idea (mainly this and this). As someone who doesn't engage with algebraic topology on a daily basis, this leaves me with a few questions.
- Are there any works that effectively highlight the significance of these spaces for the typical algebraic topologist?
- If not, could the issue possibly be attributed to an "excessive" use of filter-related terminology?
I would greatly appreciate any insights or references that could shed light on this matter.