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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.

  1. Are there any works that effectively highlight the significance of these spaces for the typical algebraic topologist?
  2. 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.

Renan Mezabarba
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    In my experience, the "typical" algebraic topologist would prefer not to think about point-set details at all, but if pressed would probably gesture at Steenrod's category of compactly generated spaces, or work with simplicial sets. – JHF Aug 02 '23 at 16:41
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    In my experience too. However, given that I have some familiarity with convergence spaces, it would be nice to be able to do something that could please them. – Renan Mezabarba Aug 03 '23 at 13:07

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