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
A geometric hyperdoctrine is a hyperdoctrine with respect to lattices that are frames.
Let $C$ be a category with finite limits. A geometric hyperdoctrine over $C$ is a functor
from the opposite category of $C$ to the category of frames, such that for every morphism $f : A \to B$ in $C$, the functor $P(A) \to P(B)$ has a left adjoint $\exists_f$ satisfying
Let $OLoc$ be the category of overt locales, and let $Frm$ be the category of frames. The functor $\mathcal{O}:OLoc^\op \to Frm$ that takes an overt locale to its frame of opens is a geometric hyperdoctrine.
Let $Set$ be the category of sets, defined as a Grothendieck topos on the singleton, and let $Frm$ be the category of frames. The subobject poset functor $\mathrm{Sub}:Set^\op \to Frm$ that takes a set to its subobject poset is a geometric hyperdoctrine.
This could be generalized to any geometric category $C$: the subobject poset functor $\mathrm{Sub}:C^\op \to Frm$ that takes an object of $C$ to its subobject poset is a geometric hyperdoctrine.
Created on November 16, 2022 at 05:25:47. See the history of this page for a list of all contributions to it.