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
symmetric monoidal (∞,1)-category of spectra
Proposed by Peter Scholze in the Analytic Stacks lectures in 2023 (Clausen-Scholze 2023) as a more well-behaved replacement for condensed sets and pyknotic sets in condensed mathematics that eliminate the size issues in the other two approaches. More specifically,
unlike condensed sets, light condensed sets form a Grothendieck topos, rather than merely a pretopos
unlike pyknotic sets, light condensed sets do not require universes
Recall that a light profinite set is a countable limit of finite sets. A light condensed set or -condensed set is a sheaf on the site of light profinite sets with finite jointly surjective families of maps as covers.
The category of light condensed sets is a Grothendieck topos.
The category of sequential topological spaces embed fully faithfully into the category of light condensed sets, and there is a functor that reflects the category of light condensed sets back onto the category of sequential topological spaces.
In the internal logic of the topos of light condensed sets, the weak limited principle of omniscience (WLPO) for the natural numbers is false (and thus excluded middle also fails).
On the other hand, the lesser limited principle of omniscience (LLPO) does hold in the topos of light condensed sets. Markov's principle also holds in the topos of light condensed sets.
The full intermediate value theorem and the Brouwer's fixed point theorem can be proven in the topos of light condensed sets.
Dustin Clausen, Peter Scholze, Analytic Stacks, YouTube playlist, website
David Wärn, On internally projective sheaves of groups (arXiv:2409.12835)
Felix Cherubini, Thierry Coquand, Freek Geerligs, Hugo Moeneclaey, A Foundation for Synthetic Stone Duality (arXiv:2412.03203)
What does the topos of (light) condensed sets classify? (MathOverflow)
Clausen–Scholze’s Theorem 9.1 of Analytic.pdf, in view of light condensed sets, AKA is the Liquid Tensor Experiment easier now? (MathOverflow)
Condensed vs pyknotic vs consequential (MathOverflow)
David Roberts, answer to Notions of convergence not corresponding to topologies (MathOverflow)
Last revised on December 6, 2024 at 23:41:27. See the history of this page for a list of all contributions to it.