topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
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.
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).
See the references at topology.
For introduction to general or point-set topology see at Introduction to Topology – 1.
Last revised on May 9, 2019 at 18:39:25. See the history of this page for a list of all contributions to it.