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
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
A kind of topological space about which one can reason “effectively”, hence constructively. Used in computable analysis, see at computable function (analysis).
Dieter Spreen, On effective topological spaces, The Journal of Symbolic Logic, Vol. 63, No. 1, Mar., 1998 (JSTOR)
Dieter Spreen, Effectively Given Spaces, Domains, and Formal Topology, 2010 PDF
See the references at computable analysis.
Discussion in relation to equilogical spaces is in
Computable Analysis and Topology_, PhD thesis CMU (2000) (pdf)
With an eye twoards application in computable physics the definition also appears as def. 2.2 in
Last revised on March 9, 2014 at 09:38:01. See the history of this page for a list of all contributions to it.