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
In algebraic topology, profinite homotopy types are frequently formed by profinite completion of a homotopy type in order to generate more ‘accessible’ information. The theory has applications both in algebraic geometry and algebraic topology.
One origin of the theory can be found in Grothendieck's Galois theory, in which he defined an algebraic fundamental group of a scheme using its finite ‘covering spaces’. These correspond to the finite field extensions in the classical case of fields, and from that perspective one can ask what the higher profinite homotopy $n$-types of a scheme should classify.
In the 1960s Artin and Mazur constructed a functor which associates to each locally noetherian scheme $X$ its étale homotopy type, $X_{et}$ , an object of $pro\Ho(SSets)$, the pro-category of the homotopy category $Ho(SSets)$ of simplicial sets. They observed that this did not correspond to a homotopy category of a model category on a category of pro simplicial sets. (This observation was the start of the search for a suitable pro-homotopy theory.)
Friedlander gave a rigidified version of the Artin-Mazur homotopy type, which he called the étale topological type? of the scheme. This was used by Quillen in the proof of the Adams conjecture, a result purely in Algebraic Topology.
Dennis Sullivan introduced profinite completions into topology in his work: D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Annals of Math., 100, (1974), 1–79. The Adams conjecture, which is a statement about purely topological phenomena, was then proved by Quillen using these ideas.
At about the same time, Bousfield and Kan, studied profinite completions amongst a wealth of other stuff in A. Bousfield and D. Kan, 1972, Homotopy limits, Completions and localizations, volume 304 of Lecture Notes in Maths , Springer-Verlag.
In the 1990s, Morel and Voevodsky defined a neat framework for the use of topological methods in algebraic geometry. They embedded the category of smooth schemes of ‘finite type over a field $k$ into a larger category of ‘$k$-spaces’, which carries the structure of a closed model category. The study of these $k$-spaces is linked to étale homotopy theory, see Schmidt, On the étale homotopy type of Morel-Voevodsky spaces, and Dan Isaksen, Etale realization on the $A^1$-homotopy theory of schemes. Adv. in Math. 184, 37–63 (2004).
A well motivated approach to profinite homotopy theory has been published by Gereon Quick, in order to give a good profinite completion construction for homotopy types; (see G. Quick, Profinite homotopy theory, Documenta Mathematica, 13, (2008), 585–612, (Archiv 0803.4082) and Some remarks on profinite completion of spaces , Arxiv preprint arXiv:1104.4659. The published version of the model category structure does not give enough generators of the fibrations, but this is corrected in a later article.
Last revised on June 10, 2019 at 20:11:45. See the history of this page for a list of all contributions to it.