[neel_k]

Do you mean topology, or geometry?

For topology, there are two main constructive approaches.

The first, better-developed one, can be found in the theory of locales. An easy intro is Vickers' Topology via Logic, from which you can level up to Johnstone's Stone Spaces.

This approach is suitable for a background logic which is constructive, but accepts the powerset axiom. If you want to restrict yourself to not just be constructive, but also predicative, then the thing to look at is Sambin's formal topology, which he surveys in his 2001 paper Some Points in Formal Topology. (https://www.math.unipd.it/~sambin/txt/SP.pdf)