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
By a condensed spectrum one should generally mean a spectrum in the context of condensed mathematics, hence a condensed object internal to spectra.
A very special case of this general notion has almost been considered in Scholze (2019), Footnote 12: The condensed simplicial abelian groups alluded to there there may be understood, under the stable Dold-Kan correspondence, to represent (the condensed version of) the particular case of connective $H\mathbb{Z}$-module spectra.
A condensed spectrum is a hypercomplete (infinity,1)-sheaf of spectra on the pro-étale (infinity,1)-site of the point, small relative to a universe $\mathcal{U}$.
A discussion of condensed spectra in the literature seems not to be available yet. For general background on condensed mathematics see:
Last revised on July 5, 2022 at 11:21:00. See the history of this page for a list of all contributions to it.