homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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 notion of vector space in condensed mathematics.
Peter Scholze, in an answer to David Roberts‘ question at Mathoverflow, says
For many (but definitely not all) applications to geometry over the real numbers, the gaseous real vector spaces work just as well, and their theory is much easier to get off the ground than liquid real vector spaces. (Roughly speaking, complex- or real-analytic spaces are fine with gaseous vector spaces, smooth manifolds not so much. The reason is that tensor products of spaces of holomorphic or real-analytic functions behave correctly under the gaseous tensor product, but tensor products of spaces of $C^\infty$-functions are only correct under the liquid tensor product.)
solid vector space?
See also at condensed mathematics
Created on September 15, 2024 at 15:20:07. See the history of this page for a list of all contributions to it.