Topology is nowadays intertwined with many other mathematical fields, like differential geometry and homological algebra, therefore yielding specialized subfields like algebraic topology, differential topology and so on. The basic study of general topological spaces (and closely related general structures like nearness spaces, uniformities, bitopological spaces and so on) remains the subject of general topology or point-set topology. It overlaps largely with set-theoretic topology, though when talking of set-theretic topology, rather than general topology, that there is a slight connotation of relevance of additional foundational axioms or other logical (say intuitionistic proofs) or set-theoretical considerations (large cardinals for example).
Some of the notions in general topology covered in the nLab include topological space, Top, Hausdorff space, specialization topology, separation axioms, sequential space, Frechet-Uryson space, compact space, Sierpinski space,
For purposes in modern mathematics sometimes roles of topological spaces are however replaced by a convenient category of topological spaces, nice topological spaces, simplicial sets, locales, sites, topoi, orbispaces, topological stacks and so on.