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
In cohesive type theory, the axiom of real cohesion states that there is a terminal Archimedean ordered field such that the shape of is a contractible type. Note that if the type theory has quotient sets, as in type theory with coequalizer types and set truncations, then the terminal Archimedean ordered field is sequentially Cauchy complete, and if the type theory is impredicative or otherwise contains the Dedekind real numbers, then is equivalent to the Dedekind real numbers and thus Dedekind complete.
This is equivalent in strength to axiom , which says that there is a terminal Archimedean ordered field such that every crisp type is discrete if and only if every function from into is a constant function. is a modality which takes types in the crisp mode to its corresponding discrete type in the cohesive mode, and a discrete type in the cohesive mode is one for which the canonical function is an equivalence of types.
Since in the terminal Archimedean ordered field, , the proofs of the equivalence of sufficient cohesion and axiom C2 also applies for real cohesion and axiom . Thus, by adapting Shulman 2018‘s proof for axiom C2 to axiom , we have that Axiom implies that every function from into a discrete type is a constant function; conversely, if every function from to a discrete type is constant, then it holds for the discrete types which are in the image of the modality. Finally, by adapting Aberlé 2024’s proof for sufficient cohesion to real cohesion, we have that the axiom of real cohesion holds if and only if every function from into a discrete type is a constant function.
The shape modality of a type is defined to be the localization of at the type
Mike Shulman, Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (arXiv:1509.07584, doi:10.1017/S0960129517000147)
Mike Shulman, Homotopy type theory: the logic of space, New Spaces in Mathematics: Formal and Conceptual Reflections, ed. Gabriel Catren and Mathieu Anel, Cambridge University Press, 2021 (arXiv:1703.03007, doi:10.1017/9781108854429)
C.B. Aberlé, Parametricity via Cohesion [arXiv:2404.03825]
Last revised on August 21, 2024 at 15:41:21. See the history of this page for a list of all contributions to it.