up: Mathematics
Some of the subfields of topology and geometry touched in the nlab: general topology, homotopy theory, rational homotopy theory, differential geometry (including symplectic geometry, synthetic differential geometry), differential topology, algebraic geometry, noncommutative algebraic geometry, noncommutative geometry.
Nowdays however there exist many frameworks in which in a unified way, the main objects (kinds of spaces, like schemes or supermanifolds) in several of the above categories can be obtained, for example ringed space, ringed site, structured (infinity,1)-topos…
Here are some nonexhaustive but longer grouped-lists of entries: geometry, topology.