Solian, A.

Institute of Mathematics AS CR, v. v. i.

Toposym 1Alexandru SolianSemi-topology of transformation groupsIn: (ed.): General Topology and its Relations to Modern Analysis and Algebra, Proceedings of thesymposium held in Prague in September 1961. Academia Publishing House of the CzechoslovakAcademy of Sciences, Prague, 1962. pp. 337--340.Persistent URL: http://dml.cz/dmlcz/700965Terms of use:© Institute of Mathematics AS CR, 1962Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czSEMI-TOPOLOGY OF TRANSFORMATЮN GROUPS A. SOLIAN Bucureşti In a previous paper [1] (see also [4], example 7), I have shown that, given any set M and a transformation group 2( of M, then between the partially-ordered set (lattice, in fact) @(M) of the equivalence relations of M and the partially-ordered set of the subgroups of 21 there can be established a dual (inverse) Galois connexion (see [2]) 35(~) and ~(S5) with ~ e g ( M ) , 35 <= 21, such that 35 -> 35(~(35)) is the closure-mapping (see [3], [4]); in other words, if 35 is a subgroup of 21, if ~(S5) is the equi­valence of M corresponding to (associated with) 35, and if 35(~(S5)) is the subgroup of 21 corresponding to (associated with) ~(35), then: 1. 35 cz 35(~(S5)); 2. 35x <= 352 ==>35(~(351)) <= S5(~(352)); 3. 35(~(S5(~(35)))) = 35(~(35)). In the present paper it will be shown that there exists a topology (in a weaker sense) of 21, which I shall call a semi-topology, such that: a) The operations of multiplication (superposition) and inversion of transforma­tions are continuous in this semi-topology (theorem 1). b) The closure of a subgroup 35 <= 21, in the sense of the above inverse Galois connexion coincides with the closure of 35 with respect to the semi-topology of 21 (theorem 2). c) If cp is a mapping of 21 onto a transformation group 21' of a set M', where (p satisfies a certain natural condition, then cp is a continuous mapping with respect to the semi-topologies of 21 and 2T (theorem 3). The present theory is not a particular case of Everett's theory [4] concerning the topology introduced in a group whose lattice of subgroups is related to another given lattice by a given Galois connexion. 1° Let M be a non-void set, 21 a transformation ^roup of M; let 35 be a subgroup of 21; then the binary relation ~(35) = ~ of M, defined by a ~ b, a, b e M <=> 3 t , T e 35 , T(O) = b is an equivalence relation of M [2], which I refer to as the equivalence associated with 35. Let ~ be an equivalence relation of M; then the subset 35(~) = 35 <= 21, defined by 35 = {T I T e 2(, T(X) ~ x, for any x e M} 22 Symposium 338 A. SOLIAN is a subgroup of 21 [2], which I refer to as the subgroup associated with ~. We have the following result [l]:1) The mappings ~ - • 25(~) and 25 —>~ (25) establish a dual (inverse) Galois connexion between (£(M), ordered by " = " where ~t — ~ 2 , ~ 1 ? ~ 2 G (J(M) <=> o (a ~ t b => a ~2 b), and the set of all subgroups of 21, ordered by inclusion; here closure is given by 23 -> 25(~(25)). 2° By a semi-topological space is meant a non-void set S of abstract elements (points) such that, for any T e S, there is given a non-void family of subsets of S (the basis of neighbourhoods of T) which satisfies the conditions: a) T belongs to all sets of its basis of neighbourhoods; b) for Tl9 T2 G S, x1 =j= T2, there exists a set belonging to the basis of neigh-bourhoods of Tl9 which does not contain T2. In the special case where S satisfies the additional condition that, for any T G S and any pair of sets Ul9 U2 in the basis of neighbourhoods of T, there exists a U3 in the same basis, for which U3 cz Ux n U2, the space S is topological space in the usual sense [5]. In a semi-topological space S, the followingterminology will be used: 1. open subset of S: any union of sets belonging to the various bases of neigh-bourhoods of points of S, or the void subset 0; 2. closed subset of S: any subset F cz S, whose complement S \ F is open; 3. neighbourhood of a point T G S: any open subset containing T; 4. closure M of a subset M c S : the set of all T G S such that [/ n M =# 0 for any neighbourhood U of T. In a semi-topological space S, we have: a) any union of open subsets is open; P) any intersection of closed subsets is closed; y) M cz M, for any M c S ; 5) a subset M is closed if and only if M = M; e) M is closed i. e. M = M, for any M c S ; £) if M t cz M2 cz S then M1 cz M2; r,) M = f) F, for any M cz S; P closed FD M 9) if M = {T} (i. e. a single-point set), then M is closed; __ ^ 0 U M J D U Mf> for any Mt, . . . , M„ cz S. i = l i = l Let S, T be two semi-topological spaces. Consider the cardinal product of the sets S, T as point set; as basis of neighbourhoods of a point (tr, T), CT G S, T G T, take the family of all pairs (U, V) where U and V belong to the basis of neighbourhoods of o 1) See also [4] and [2] (where it is proved that the corresponding mappings are monotone). A. SOLIAN 339 in S and of T in T, respectively. Thus we obtain a semi-topological space S x T, which shall be called the cartesian product of the given spaces S, T. Let S, S' be two semi-topological spaces; a uniform mapping / : 5 -* S' is by definition a continuous mapping of S into S' if for any x e S, and any neighbourhood *U' off(x) in S', one can find a neighbourhood 17 of T in S with/(U) c U'. A semi-topological group is by definition a non-void set © of abstract elements such that following conditions are fulfilled: I. © is a group with respect to a certain law of composition, denoted by "." or by juxtaposition. II. © is a semi-topological space. III. The mappings p : © x © -> © and i : © ~> © defined by p(cr, T) = cr. T, i(c) = a"1 for a, T e © are continuous. 3° Let M be a non-void set and 21 a transformation group of M. As law of composition in 2( take the superposition of transformations. Let T e 21; as basis of neighbourhoods of T in 21 take the family {U*}, xeM, where Wx = {a | a e % a(x) = x(x)} . Then we have the following results: Theorem 1. With respect to the defined operation and basis of neighbourhoods of a point, 21 is a semi-topological group. Theorem 2. If 35 is a subgroup of 21, then 2$ = 35(~(35)) (where by 35 we mean the closure in the sense of the semi-topology in 21), i. e. the closure of a subgroup in the Galois connexion coincides with its closure in the semi-topology of 21. Theorem 3. Let M, M' be non-void sets, 21, 21' transformation groups of M, M\ respectively; letfresp. cp be mappings of M onto M', resp. o/2l onto 2T, satisfying the condition f(r(x)) = ((p(x))(f(x)), for any xeM, % e 21 ; then the mapping cp : 21 -> 21' is a group homomorphism and a continuous mapping of 21 onto 21' (in the semi-topology just defined). We mention also the following p r o p e r t i e s : / / 21 acs regularly on M (i. e. T^XQ) = t2(x0) for some x0e M implies xx = T 2 whenever T1? T2 e 21) then the semUtopology of 21 is discrete (i. e. any x e 21 has a single-point neighbourhood {T}). A subgroup 35 c X(M) (where X(M) is the group of all transformations of M) is dense in X(M) (i. e. 35 = X(M), where the closure 35 is taken with respect to the semi-topology of X(M)) if and only if it acts transitively on M. 22* 340 A. SOLIAN References [1] A. Solian: Equivalence Relations and Associated Subgroups. Paper read at the Ilnd Hungarian Mathematical Congress, Budapest, August 1960. [2] P. Dubreil: Algebre. Heme edition, Gauthier-Villars, Paris 1954. [3] O. Ore: Galois Connexions. Trans. Amer. Math, Soc, vol. 55 (1944), 493-513. [4] C. J. Everett: Closure Operators and Galois Theory in Lattices. Ibid., vol 55 (1944), 514-525. [5] JI. C. TloHmpmuH: HenpepbiBHbie rpynriM. THTTJI, MocKBa 1954. 

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Semi-topology of transformation groups

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Semi-topology of transformation groups

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Institute of Mathematics AS CR, v. v. i.

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