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
Due to Paul Taylor, Abstract Stone Duality (ASD) is a re-axiomatisation of the notions of topological space and continuous function in general topology in terms of a lambda-calculus of computable continuous functions and predicates.
Abstract Stone duality is both constructive and computable, thus being one approach to synthetic topology.
The topology on a space is treated not as a discrete lattice, but as an exponential object of the same category as the original space, with an associated λ-calculus (which includes an internal lattice structure). Every expression in the λ-calculus denotes both a continuous function and a program. ASD does not use the category of sets (or any topos), but the full subcategory of overt discrete objects plays this role (an overt object is the dual to a compact object), forming an arithmetic universe (a pretopos with lists) with general recursion; an optional ‘underlying set’ axiom (which is not predicative) will make this a topos.
The classical (but not constructive) theory of locally compact sober topological spaces is a model of ASD, as is the theory of locally compact locales over any topos (even constructively). In “Beyond Local Compactness” on the ASD website, Taylor removes the restriction of local compactness.
Paul Taylor, Abstract Stone Duality (www.paultaylor.eu/ASD)
Paul Taylor, Review of Abstract Stone Duality (2004) [pdf]
On Dedekind real numbers via abstract Stone duality:
Paul Taylor, Dedekind cuts (2007-2009?)
Andrej Bauer, Paul Taylor, The Dedekind reals in abstract Stone duality, Mathematical Structures in Computer Science 19 4 (2009) 757-838 [doi:10.1017/S0960129509007695]
review in:
See also:
Last revised on February 21, 2023 at 11:01:11. See the history of this page for a list of all contributions to it.