The other commenters are making some very good points about how fundamental dimensionality is in studying vector spaces. Just wanted to add that if you'd like

The other commenters are making some very good points about how fundamental dimensionality is in studying vector spaces. Just wanted to add that if you'd like to take a more set-theoretic approach to the topic of dimensionality, you might be interested in Matroid theory, which is a generalization of the notion of linear independence in linear algebra. I found this video to be helpful in getting a basic sense for what Matroids are: https://www.youtube.com/watch?v=FKTPFy-MGp0 https://www.youtube.com/watch?v=FKTPFy-MGp0