Whenever we are given two topological groups, we insist that a homomorphism between them be continuous. The diagonal embedding embeds z as a dense subgroup of z. An interesting question is whether two 3manifolds can be distinguished by computing and comparing their collections of finite covers. It was not known whether the profinite completion functor on prosimplicial sets admits a right adjoint or not cf. The most popular way to define a topological space is in terms of open sets, analogous to those of euclidean space. Ix of fuzzy sets is called a fuzzy topology for xif it satis es the following three axioms. A topological group gis a group that is also a topological space such that the multiplication map m. An explanation of why they are different, only found recently, is by considering a more refined notion of a classifying space, which is a uniform space welldefined up to uniform homotopy equivalence see, for lack of a better account, subsection 1. Equivalence of definitions of profinite group groupprops. Ivanov ws 201920 algebraic geometry i christmas exercise. The profinite topology on a group is a topology on the underlying set of the group defined in the following equivalent ways. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Topological space definition of topological space by. The following are equivalent for a topological group.
A profinite group is a hausdorff, compact, and totally disconnected topological group. Metricandtopologicalspaces university of cambridge. Then we call k k a norm and say that v,k k is a normed vector space. One checks that a discrete topological space is totally disconnected, and that a product of totally disconnected spaces is totally disconnected, and that being totally disconnected is hereditary. A pro nite space group is the projective limit of nite sets groups. A collection of topological ramsey spaces of trees and. For any ring r, direct limits exist in the category rmod of rmodules. Moreover, for each profinite group g and each element g. Galois theory o ers a natural frame in order to describe galois groups as pro nite groups. Then every sequence y converges to every point of y. Topological space definition is a set with a collection of subsets satisfying the conditions that both the empty set and the set itself belong to the collection, the union of any number of the subsets is also an element of the collection, and the intersection of any finite number of the subsets is an element of the collection. This is not the most convenient characterization of a profinite space. In mathematics, more precisely in formal language theory, the profinite words are a generalization of the notion of finite words into a complete topological space. These and several other studies have many similarities in results.
We then looked at some of the most basic definitions and properties of pseudometric spaces. This equivalence carries over to profinite vs stonetopological algebras for a number of algebraic theories, such as groups, monoids, semigroups, and rings. Nov 08, 2014 as i explained in a previous answer qiaochu yuans answer to what is a topological space. It has as a basis of open subsets all left cosets of subgroups of finite index it has as a basis of open subsets all right cosets of subgroups of finite index it has as a basis of open subsets all cosets of normal subgroups of finite index.
As i explained in a previous answer qiaochu yuans answer to what is a topological space. It is based on the halpernlauchli theorem, but different from the milliken space of strong subtrees. The pro nite completion of the integers, the padic. As in 12, we denote by rxthe category of coverings of x, i. Of course, one could have defined them using the etale topological types of hypercovers, which are simplicial algebraic space, of algebraic stacks. Continuous cohomology and homology of profinite groups 3 proof.
A subset of an ideal topological space is said to be closed if it is a complement of an open set. Katonakayama spaces, infinite root stacks, and the. What is an intuitive explanation of a profinite topology. For a log scheme locally of finite type over c, a natural candidate for its profinite homotopy type is the profinite completion of its katonakayama space. There are also plenty of examples, involving spaces of functions on various domains. In the notion of a topological vector space, there is a very nice interplay between the algebraic structure of a vector space and a topology on the space, basically so that the vector space operations are continuous mappings. Connectedness is one of the principal topological properties that are used to distinguish topological spaces a subset of a topological space x is a connected set if it is a connected.
Let fr igbe a sequence in yand let rbe any element of y. Preliminaries in this section we recall the basic topological notions that are used in the paper. A topological space is profinite if it is homeomorphic to a limit of a diagram of finite discrete spaces. Direct limits, inverse limits, and profinite groups math 519 the rst three sections of these notes are compiled from l, sections i. Profinite spaces, connected components, coarser topology, re.
It is a compact totally disconnected t0 topological group. This is further complicated by the question of whether or. It turns out that a great deal of what can be proven for. A topological space is an aspace if the set u is closed under arbitrary intersections. Continuous cohomology and homology of profinite groups. Profinite rigidity for seifert fibre spaces springerlink. The effective properties of these topological algebras have been studied independently, for example. Beware there are two possible interpretations of this term. In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Profinite homotopy theory 3 may identify f with a full subcategory of e.
Chapter 9 the topology of metric spaces uci mathematics. Classifying spaces of a profinite groups mathoverflow. Introduction in chapter i we looked at properties of sets, and in chapter ii we added some additional structure to a set a distance function to create a pseudomet. A new result on profinite completions will be given in the course of proving the comparison theorem. Topological spaces can be fine or coarse, connected or disconnected, have few or many dimensions. T limiti is the inverse limit of finite discrete topological spaces ti. If v,k k is a normed vector space, then the condition du,v ku. Thus, z is the free profinite group generated by one element. Browse other questions tagged abstractalgebra generaltopology grouptheory topologicalgroups profinitegroups or ask your own question.
In this paper, we solve this question completely for closed orientable seifert fibre spaces. For a subset v of x, let clv and intv denote the closure and the interior of v, respectively, with respect to the topological spacex. It is the inverse limit of an inverse system of finite groups, each equipped with the discrete topology. Given this definition, it is possible to recover the first definition using the inverse limit lim topological group gis a group that is also a topological space such that the multiplication map m. Profinite rigidity for seifert fibre spaces pdf paperity. The entry linked to here treats another more purely topological concept.
Just as the term space is used by some schools of algebraic topologists as a synonym for simplicial set, so profinite space is sometimes used as meaning a simplicial object in the category of compact and totally disconnected topological spaces, i. A limit of such a system is a fibration whose fibres are profinite topological groups. On the category of profinite spaces as a reflective. It has as a basis of open subsets all left cosets of subgroups of finite index. In particular, all seifert fibre spaces are distinguished. Topologytopological spaces wikibooks, open books for an. A topological group is a group g which is also a topological space with the property that the multiplication map m. Here and in general for profinite groups, whenever we use the term. An internal group in the category of stone spaces profinite spaces and. If fis a presheaf on the topological space x, and x2x, consider the collection u x of open neighborhoods of x.
Request pdf compact homeomorphism groups are profinite if the homeomorphism group hx of a tychonoff space x is compact in the compact open topology, then it. However, the homotopy theory in this paper gives an intrinsic definition of topological types of algebraic stacksnot depending on hypercoversin such a way that the topological types of. We give an example of its application by proving a partition theorem for profinite graphs. One defines interior of the set as the largest open set contained in. And a profinite language is a set of profinite words. A collection of topological ramsey spaces of trees and their. Some new sets and topologies in ideal topological spaces. In this section we study profinite topological algebras. This is not the most convenient characterization of a profinite. If uis a neighborhood of rthen u y, so it is trivial that r i. In particular, direct limits exist in the category ab of abelian groups. A topological group satisfying both equivalent conditions is termed a profinite group. Any normed vector space can be made into a metric space in a natural way. Pdf representing a profinite group as the homeomorphism.
Geom dedicata profinite rigidity for seifert fibre spaces gareth wilkes 0 0 mathematical institute, university of oxford, andrew wiles building, radcliffe observatory quarter, woodstock road, oxford ox2 6gg, uk an interesting question is whether two 3manifolds can be distinguished by computing and comparing their collections of finite covers. Topological types of algebraic stacks international. Request pdf compact homeomorphism groups are profinite if the homeomorphism group hx of a tychonoff space x is compact in the compact open topology, then it is a profinite topological group. Informally, 3 and 4 say, respectively, that cis closed under. Finally, for a log scheme not necessarily over c, another natural candidate is the profinite \etale homotopy type. We wish to show that a group gis pro nite if and only if it is a compact, hausdor, and totally disconnected topological group.
Introduction to topological groups dikran dikranjan to the memory of ivan prodanov abstract these notes provide a brief introduction to topological groups with a special emphasis on pontryaginvan kampens duality theorem for locally compact abelian groups. Compact homeomorphism groups are profinite request pdf. This particular topology is said to be induced by the metric. We treat g k as a fundamental object of study because it allows us to control all separable extensions l of k in one stroke. A profinite space is a compact hausdorff and totally disconnected topological space. The properties of the topological space depend on the number of subsets and the ways in which these sets overlap. Katonakayama spaces, infinite root stacks, and the profinite.
This notion allows the use of topology to study languages and finite semigroups. We show that a topological algebra is profinite if and only if its carrier space is a stone space and its translation monoid is equicontinuous theorem 4. A topological space is profinite if it is homeomorphic to the inverse limit of an inverse system of finite topological spaces. We construct a collection of new topological ramsey spaces of trees. For example, profinite words are used to give an alternative characterization of the algebraic notion of a variety of finite semigroups.