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
The axiom of punctual cohesion or axiom C1 states that there is a pointed type with designated point 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.
Shulman 2018 showed that axiom C1 implies axiom C0, which 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.
The boolean domain is discrete.
Theorem 6.19 of Shulman 18 says that the unit type is crisply discrete, and theorem 6.21 of Shulman 18 says that the sum type of two crisply discrete types is itself crisply discrete. Since the boolean domain is the sum type of two copies of the unit type, the boolean domain is crisply discrete, thus discrete.
Let be a cohesive type with a tight apartness relation. Assuming punctual cohesion and crisp excluded middle, if the tight apartness relation on is decidable, then is discrete.
Since the tight apartness relation and equality are incompatible propositions, we have for all and that
so we can define functions by induction on the boolean domain. Given a function there is a function defined as
Since the boolean domain is discrete, is constant, but , so and for all , meaning that is also a constant function. Thus is discrete.
The natural numbers are discrete.
The tight apartness relation on is given by denial inequality, and denial inequality is decidable since equality is decidable in , implying that is discrete.
Assuming crisp excluded middle and punctual cohesion, types with decidable tight apartness relations are closed under function types in the cohesive mode.
Given types and with decidable tight apartness relations, types and are discrete, which means that the function type is also discrete by axiom C0, so we may assume that any functions and are crisp. Then the tight apartness relation on defined by
is crisp, so by crisp excluded middle, we have or . But is just , so we are done.
Assuming crisp excluded middle and punctual cohesion, the Cauchy real numbers , HoTT book real numbers , the -Dedekind real numbers constructed using Dedekind cuts valued in the initial -frame , and the flat modality of the terminal Archimedean ordered field are all isomorphic Archimedean ordered fields in the cohesive mode.
Since and both have decidable tight apartness relations, the function type has a decidable tight apartness relation, which implies the limited principle of omniscience.
The limited principle of omniscience implies that for each rational number , the left and right Dedekind cuts for an element of , evaluated at , and , are both decidable propositions. In addition, limited principle of omniscience implies that any existential quantification of a decidable predicate over the rational numbers is itself a decidable proposition. The pseudo-order relation on is defined as
Since and are both decidable, the conjunction is also decidable, and so is the existential quantifier by LPO for the natural numbers, and by definition the strict order on , which is the analytic LPO for .
The analytic LPO for implies that every element in has a locator by corollary 11.4.3 of the HoTT book. This then implies that is isomorphic to by corollary 11.4.1 of the HoTT book, and since the Cantor-Schroeder-Bernstein theorem holds for Archimedean ordered fields and ring homomorphisms in the category of Archimedean ordered fields, is also isomorphic to both and .
Finally, every element of can be assumed to be a crisp element, and crisp excluded middle implies that every element of has a locator, and is thus a crisp Cauchy real number. Thus, we have since already embeds into , and since has decidable tight apartness, it is discrete and we have . Thus, we have in the cohesive mode
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 19:55:58. See the history of this page for a list of all contributions to it.