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
Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
symmetric monoidal (∞,1)-category of spectra
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
Dustin Clausen says in this comment on the nCafé:
Namely, I claim that if you’re willing to pass to suspension spectra, then you do get a fully well-defined functor which takes a CW-complex (viewed as a condensed set) to the suspension spectrum of its associated anima. But now this is not a functor to ordinary spectra, but a functor to condensed spectra which happens to take “discrete” values (discrete in the condensed direction) on CW-complexes.
Actually, what you really have is a colimit-preserving endofunctor of condensed spectra, which when applied to the suspension spectrum of a CW complex (viewed as a condensed set) gives the suspension spectrum of its associated anima. This is the “solidification” functor, which works for spectra much in the same way it works for abelian groups.
Created on May 30, 2022 at 13:10:10. See the history of this page for a list of all contributions to it.