nLab
general topology

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-theoretic 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).

For the purposes of the nLab, the phrase point-set topology may conveniently signify that the traditional conception of spaces as consisting of points is being utilized. As opposed to doing “pointless” topology, where spaces are conceived as consisting of open sets, as in the theory of locales.

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.

Revised on December 31, 2016 10:32:55 by Todd Trimble (67.81.95.215)