CC BY
9
A contribution to the theory of continuous homomorphisms
Kasymova T.
К теории непрерывных гомоморфизмов Касымова Т. Дж.
Касымова Тумар Джапашевна / Kasymova Tumar Japashevna - кандидат физико-математических наук, доцент, кафедра алгебры, геометрии и топологии, факультет математики, информатики и кибернетики,
Кыргызский национальный университет им. Ж. Баласагына, г. Бишкек, Кыргызская Республика
Abstract: for continuous homomorphisms of topological groups a concept of base has been introduced and the parallel-ability (inclusion) of continuous homomorphisms has been researched by means of it in the category GTOP(H).
Аннотация: для непрерывных гомоморфизмов топологических групп введено понятие базы, посредством которого исследована параллельность (вложение) непрерывных гомоморфизмов в категории GTOP(H).
Keywords: a base of continuous homomorphism, category, character.
Ключевые слова: база непрерывного гомоморфизма, категория, характер.
УДК 515.12
An idea to transfer some concepts and statements concerning spaces to mappings allows to generalize many results. In this way works of L. S. Pontryagin [7], B. A. Pasynkov [6], A. A. Borubaev [1], А. А. Chekeev [2] and others are known. A transferring of these results to algebraic objects and studying of their behavior is the actual task significantly enriching the theory of uniform spaces and as a result the theory of uniformly continuous mappings. So, for example, the topological group has algebraic structure, on the one hand, and is a topological space on another one. Below the concept of base of uniformly continuous mapping [1] is postponed for continuous homomorphisms of topological groups.
Remind of some basic concepts from books [5], [3].
A family B ( x) of neighborhoods of X is called a base for a topological space X at the point X or a local base if for any neighborhood V of X there exists a U e B (x) such that x e U с V. If B is a base for X then the family B (x) consisting of all elements of B that contain X is a base for X at the point X . On the other hand, if for every x e X a base for X at the point X is given then the union B = kj{B (x) : x e X} is a base for X .
The character of a point x e X is defined as the smallest cardinal number of form |B (x)| , where B (x) is a
base for a topological space X at the point X ; |-| stands for cardinality; this cardinal number is denoted by %(x, X). The character of a topological space X is defined as the supremum of all numbers %(x, X) for x e X , i.e. %(X) = sup{%(x, X): x e X}.
Definition [3]. A set G allocated with structures of group and topology is called a topological group if it satisfies the following two axioms:
(GTi) Mapping (x, у) l—> xy of product GxG into G is continuous.
(GTn) Mapping x I—> x 1 of group G into itself (symmetry of group G) is continuous.
Axioms (GT,) and (GTn) are equivalent to the next axiom:
(GT) Mapping (x, у) I—> xy 1 of product GxG into G is continuous.
Every topological group has a base B (e) of neighborhoods filters of unit, satisfying to the following axioms: (GVI) For any U e B (e) there exists V e B (e) such that VV с U.
(GVII) For any U e B (e) there exists V e B (e) such that Vс U.
(GVnI) For any a e G and U e B (e) there exists V e B (e) is containing into aUa 1.
Axioms (GVI) and (GVII) can be reformulated as:
(GV) For any U e B (e) there exists V e B (e) such that VV-1 с U.
Let f : G ^ H be a continuous homomorphism of topological group G into topological group H, B (eG),
B (e) are a bases of neighborhoods filters of units eG e G and e e H, respectively, Bf (eG) be a
neighborhoods system, generated group topology on G, generally speaking, more weak, than initial one, i.e.
Bf (eG ) C B (eG ) .
Definition 1. A neighborhoods system Bf (eG ) is said to be a base of continuous homomorphism f , if for any neighborhood U 6 В (ea) there exist such neighborhoods V 6B (e) and W 6 By (eG) that
f(V) W c U holds and a character % (f) < т, if |®y (eG )| <T .
It is known [4], that objects in category GTOP are all separated topological groups, and morphisms are continuous homomorphisms.
We denote a category of all continuous homomorphisms as GTOP(H) it’s an objects are continuous homomorphisms f : G ^ H, g: N ^ H with fixed bases By (eG) and B (eN) respectively, and morphism from an object f 6 GTOP(H) into object g 6 GTOP(H) is called homomorphism h : G ^ N continuous with respect to topologies, induced by bases Bf (eG ) and B (eN ) such that f = g ■ h .
In this case we write h : f ^ g .
Lemma. If Bf (eG) is a base of object f: G ^ H of category GTOP(H), then a family N = П | W: W e ®y (eG ) j is a normal subgroup of group G .
Proof. Let G be a topological group having a base B (eG) of neighborhoods filters of unit eG 6 G, Bf (eG) be a neighborhoods system, generated group topology on G, generally speaking, more weak, than initial one, i.e. ®y(eG)c®(eG). Denote as N = П{W: W e <bf (eG)j a intersection of all neighborhoods of the base Bf (eG ) of continuous homomorphism f : G ^ H (according to Definition 1).
We show that N is a subgroup of group G . By construction N c W for any W 6 By (eG ) . On axioms (GV) of base of filter of unit and (GT) of topological group we have:
1) for any W 6 By (ea) there exists such V 6 By (eG) that VV 1 c W, moreover NN_1 c W for any
W 6 Bf (eG), it means NN_1 c N, i.e. N is a subgroup of group G .
2) For any W 6 By (eG) and a 6 N there exists such V 6 By (eG) is containing into aWa~', i.e.
V c aWa 1 and N c aWa 1. Then aT^Na c W, hence aNaTy c N, i.e. N is a normal subgroup of group
G.
The concept of the mapping parallel to space was introduced by B. A. Pasynkov [6] for topological spaces, it was done by A. A. Borubaev [1] for uniform spaces.
We introduce this concept for continuous homomorphisms of topological groups.
Definition 2. Let f : G ^ H be a continuous homomorphism of topological group G into topological group
H, ж : G ^ G/jq be a projection of G onto factorgroup on the normal subgroup N . A continuous
homomorphism f is parallel to projection ж (is denoted as f ||ж) if there exists such continuous mapping i: G ^ G/fyx H of topological group G into topological group G/fyx H, that i = f Аж and f = pH | : G ^ H. So the mapping i is called inclusion.
Theorem 1. Let f 6 GTOP(H), where f : G ^ H is continuous homomorphism - an object of category GTOP(H), %(f) < т. Then there exists such normal subgroup N c G that f ||ря , where
Ph : % x H ^ H.
Proof. Let В * (eG ) be a base of continuous homomorphism f: G ^ H - an object of category GTOP(H), and
В
G.
(eG )| <T . There exists natural homomorphism n: G ^ G/n is a projection of group G onto factorgroup
on normal subgroup N = fl{W:We(Bf(eG)} (according to Lemma), given on mle 7г(х) = xN for any xeG.
Let us consider diagonal product i = f An. It is continuous as diagonal product of continuous homomorphisms. We prove, that f An is inclusion of continuous homomorphism f into projection pH . Note,
that f = Ph ■ (f An).
We show, that f An is one-to-one. Let x, X e G and x ^ x2. Пусть f (x) e V., V. eB (e) , i = 1,2
of group H. Then there exist W, E B f (eG) such that x. E ( f лтг) 1 ( W; )сЦ., / = 1,2. We have
((/A^CWO nf'CV,)) o/-4V2)) = 0, hence
(Wi x Vi) о(W2 x V2) = 0.
Then, on properties of neighborhoods system [5], U c Uj о U2 eB (eG ) and V c V о V2 eB (e) such that (U о f ^(V))!) о (U о f _1(V))(x2 ) = 0. Therefore, there exists
W c W о W e Bj- (eG) ^ В (eG) such that (f An)_1(W) c U. It means that (f An)(x) Ф (f An)(x2)
, i.e. fAn is isomorphism.
As topological group G is isomorphically enclosed into G/^ x H, then (f An)(U) e W x V|(^)(G) for any neighborhood UeB(eG). Then on Definition 1 Wо f :(V) cU, hence
(W x V) о (f An)(G) c (f An)(U). The projectionря : x H ^ H of Cartesian product of and
topological group H on factor H is given by (n(x), f (x)) = ря (x) for any x e G. So the diagonal product f An is inclusion of f into pH , i.e. /||ря .
References
1. Borubaev A. A. Uniform spaces and uniformly continuous mappings [Text] /А. А. Borubaev. - Bishkek, 1990. 170 p. (in Russian).
2. Borubaev А. А. Uniform structures on the topological spaces and groups [Text] /А. А. Borubaev, А. А. Chekeev. - KSPU named after I. Arabaev, Bishkek, 1997. - 267 p. (in Russian).
3. Bourbaki N. General topology. Topological groups. [Text] / N. Bourbaki. - Moscow: Nauka, 1969. - 392 p. (in Russian).
4. Bucur I. Introduction to the theory of categories and functors. [Text] / I. Bucur, A. Deleanu. - Moscow: Mir, 1972, 259 p. (in Russian).
5. EngelkingR. General topology. [Text] / R. Engelking. - Moscow: Mir, 1986, 515 p. (in Russian).
6. Pasynkov B. A. On transferring on mappings of some concepts and statements concerning spaces. [Text] / B. A. Pasynkov // Mappings and functors. - Moscow: MSU, 1984. - P. 72-102. (in Russian).
7. Pontryagin L. S. Continuous groups. [Text] / L. S. Pontryagin. - Moscow: Fizmatgiz, 1973. - 527 p. (in Russian).
8. Kasymova T. J. To the theory of continuous homomorphisms [Text] / T. J. Kasymova // Abstracts of the Issyk-Kul International Mathematical Forum / ed. by Acad. A. Borubaev. - Bishkek: Kyrgyz Mathematical Society, 2015. - P. 23.
