natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
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
Synthetic topology, like synthetic domain theory, synthetic differential geometry, and synthetic computability, are part of synthetic mathematics. It uses the internal logic or type theory of a topos or -topos to develop topology. This is closely related to topology via logic and abstract Stone duality. One example of synthetic topology is synthetic Stone duality.
Synthetic topology has semantics in many categories, and in particular in many categories of “spaces”. Thus types may be regarded not just as sets but as topological objects. Interestingly, a good deal of this “topology” can be detected intrinsically in type theory, often corresponding to the possible failure of principles of classical mathematics such as excluded middle, via axioms such as Phoa's principle or synthetic quasi-coherence for a dominance.
Martín Escardó, Synthetic topology of data types and classical spaces, Electronic Notes in Theoretical Computer Science (ENTCS), Volume 87, Pages 21 - 156, 01 November 2004 [doi:10.1016/j.entcs.2004.09.017, pdf]
Andrej Bauer, Davorin Lešnik, Metric Spaces in Synthetic Topology, Annals of Pure and Applied Logic, Volume 163, Issue 2, February 2012, Pages 87-100 [doi:10.1016/j.apal.2011.06.017, pdf]
Davorin Lešnik, Unified Approach to Real Numbers in Various Mathematical Settings, 2014 [arXiv:1402.6645]
Martin E. Bidlingmaier, Florian Faissole, Bas Spitters, Synthetic topology in Homotopy Type Theory for probabilistic programming. Mathematical Structures in Computer Science, 2021;31(10):1301-1329. [doi:10.1017/S0960129521000165, arXiv:1912.07339]
Davorin Lešnik, Synthetic Topology and Constructive Metric Spaces [arXiv:2104.10399]
Davorin Lešnik, “Synthetic Topology”. In: Douglas Bridges, Hajime Ishihara, Michael Rathjen, Helmut Schwichtenberg, editors, Handbook of Constructive Mathematics, Encyclopedia of Mathematics and its Applications, Cambridge University Press, pp. 445 - 482, 04 May 2023. [doi:10.1017/9781009039888.018]
Felix Cherubini, Thierry Coquand, Freek Geerligs, Hugo Moeneclaey, A Foundation for Synthetic Stone Duality [arXiv:2412.03203]
Jonathan Sterling, Lingyuan Ye, Domains and Classifying Topoi [arXiv:2505.13096]
Fredrik Bakke, Jonathan Sterling, Mark Damuni Williams, Lingyuan Ye, The Synthetic Sierpiński Cone [arXiv:2605.00773]
Last revised on June 24, 2026 at 22:50:47. See the history of this page for a list of all contributions to it.