constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
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
Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
Let be a 01-bounded semilattice, that is, a semilattice with an absorbing element. Phoa’s principle states that every endofunction is monotonic and the precomposition function by the embedding of the booleans into is an embedding.
If is a distributive lattice, then Phoa’s principle is equivalent to the linear interpolation condition that for all endofunctions and elements :
In Pos the category of posets and monotonic functions, Phoa’s principle holds for the boolean domain . In synthetic Stone duality, Phoa’s principle holds for the type of open propositions . In synthetic domain theory, Phoa’s principle holds for a dominance .
Phoa’s principle is named after Wesley Phoa.
Wesley Phoa: Domain theory in realizability toposes, PhD thesis, Trinity College, Cambridge (November 1990) [lfcs:91/ECS-LFCS-91-171, pdf]
Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz, Directed univalence in simplicial homotopy type theory (arXiv:2407.09146)
Thierry Coquand, Freek Geerligs, Hugo Moeneclaey, Directed Univalence in Synthetic Stone Duality, unfinished draft; LaTeX documents can be found here.
Jonathan Sterling, Baby steps in higher domain theory, talk at Homotopy Type Theory and Computing – Classical and Quantum, Center for Quantum and Topological Systems (video)
Leoni Pugh, Jonathan Sterling, When is the partial map classifier a Sierpiński cone? (arXiv:2504.06789)
Jonathan Sterling, Lingyuan Ye, Domains and Classifying Topoi (arXiv:2505.13096)
Last revised on June 22, 2025 at 15:46:37. See the history of this page for a list of all contributions to it.