nLab general topology

Idea

What is called general topology or point-set topology is the study of the basic concepts of topology based on the concept of topological spaces considered as sets (of points) equipped with a topology. The term is to contrast with other areas of topology, such as algebraic topology or differential topology, and specifically to contrast with homotopy theory, where only the (weak) homotopy type of a topological space matters, not the homeomorphism type of its underlying topologized point-set, and where other models/presentations for these homotopy types may be used (other than “topological point-sets”), notably simplicial sets.

The study of generalizations of topological spaces in the guise of sets with extra structure, such as to nearness spaces, uniformities, bitopological spaces and so on, may still be regarded as the subject of “point-set topology”.

There is also the point-less formulation of generalized topological spaces in terms of locales. This might still be regarded as part of general topology, but it is manifestly not to be counted as “point-set topology”, and is known insteasd as pointfree topology (or pointless topology).

Other non-point-set approaches include formal topology and abstract Stone duality.

There is also the term set-theoretic topology, but that tends to allude to additional foundational axioms being considered or other logical (say intuitionistic) or set-theoretical considerations (large cardinals for example).

Related entries

• Basic introduction to general topology
• geometry, higher geometry

References

• Nicolas Bourbaki, General topology, Elements of Mathematics, Springer (1971, 1990, 1995) [doi:10.1007/978-3-642-61701-0]

For more see the references at topology.

For introduction to general or point-set topology see at Introduction to Topology – 1.