On s-Topological Groups

In this paper we study the class of s-topological groups and a wider class of S-topological groups which are defined by using semi-open sets and semi-continuity introduced by N. Levine. It is shown that these groups form a generalization of topological groups, and that they are different from several distinct notions of semitopological groups which appear in the literature. Counterexamples are given to strengthen these concepts. Some basic results and applications of sand S-topological groups are presented. Similarities with and differences from topological groups are investigated.


Introduction
If a set is endowed with algebraic and topological structures, then it is natural to consider and investigate interplay between these two structures. The most natural way for such a study is to require algebraic operations to be continuous. It is the case in investigation of topological groups: the multiplication mapping and the inverse mapping are continuous. Similar situation is with topological rings, topological vector spaces and so on. However, it is also natural to see what will happen if some of algebraic operations satisfy certain weaker forms of continuity. Such a study in connection with topological groups started in the 1930s and 1950s and led to investigation of semitopological groups (the multiplication mapping is separately continuous), paratopological groups (the multiplication is jointly continuous), quasi-topological groups (which are semitopological groups with continuous inverse mapping). It is naturally suggested to identify conditions under which these classes of groups are topological groups. In the last twenty-thirty years many nice results related to this mathematical discipline appeared in the literature, and the list of such papers is too long to be mentioned here; because of that we refer the reader to two excellent sources: the monograph [2] by Arhangel'skii and Tkachenko, and Tkachenko's survey paper [21], and references therein.
Our approach in the present paper is different, and we require less restrictive conditions on the group operations: neither of the operation is required to be continuous. Our assumption is that the group operations are semicontinuous in the sense of N. Levine [16] (equivalently, quasi-continuous in the sense of Kempisty [13]). In this way we define two classes of groups (together with a topology) called here s-topological groups and S-topological groups. (Notice that the notion of s-topological groups have already appeared in the literature [3].) Some basic results on these generalizations of topological groups are obtained, and similarities with and differences from topological groups are explored.

Preliminaries
Throughout this paper X and Y are always topological spaces on which no separation axioms are assumed. For a subset A of a space X the symbols Int(A) and Cl(A) are used to denote the interior of A and the closure of A. If f : X → Y is a mapping between topological spaces X and Y and B is a subset of Y , then f ← (B) denotes the pre-image of B. Our other topological notation and terminology are standard as in [10]. If (G, * ) is a group, then e denotes its identity element, and for a given x ∈ G, x : G → G, y → x * y, and r x : G → G, y → y * x, denote the left and the right translation by x, respectively. The operation * we call the multiplication mapping m : G × G → G, and the inverse operation x → x −1 is denoted by i.
In 1963, N. Levine [16] defined semi-open sets in topological spaces. Since then many mathematicians explored different concepts and generalized them by using semi-open sets (see [1,8,11,19,20] [16], and of semi-closed sets and the semi-closure in [6,7]. Recall that a set U ⊂ X is a semi-neighbourhood of a point x ∈ X if there exists A ∈ SO(X) such that x ∈ A ⊂ U . A set A ⊂ X is semi-open in X if and only if A is a semi-neighbourhood of each of its points. If a semi-neighbourhood U of a point x is a semi-open set we say that U is a semi-open neighbourhood of x.
Clearly, continuity implies semi-continuity; the converse need not be true.
Notice that a mapping f : X → Y is semi-continuous if and only if for each In [13], Kempisty defined quasi-continuous mappings: a mapping f : X → Y is said to be quasi-continuous at a point x ∈ X if for each neighbourhood [17]). Neubrunnová in [18] proved that semi-continuity and quasi-continuity coincide.

s-Topological Groups
In this section, the notion of S-topological groups is introduced by using semi-open sets and semi-continuity of the group operations. Relations between this class of groups and other classes of groups endowed with a topology are considered. It is pertinent to mention that this notion of Stopological groups is different from the notion of semi-topological groups already available in the literature, in particular from semi-topological groups introduced in [3] and called here s-topological groups. (a) A triple (G, * , τ ) is said to be an S-topological group if (G, * ) is a group, (G, τ ) is a topological space, and (a) the multiplication mapping m : It follows from the definition that every topological group is both an s-and S-topological group. It is shown in [3, Theorem 7] that every stopological group is an S-topological group. The examples below show that the converses are not true.
Remark 3.1. In the literature there are several different notions of semitopological groups [4] (the multiplication mapping is continuous in each variable separately and the inverse mapping is continuous), [2,5,12,21] (the multiplication mapping is continuous in each variable separately; throughout this paper we adopt this definition of semitopological groups), [3] (see the above definition). This fact motivated us to use the name S-topological groups for the introduced class and so to avoid a possible confusion. Our groups are different from the other mentioned groups. The Sorgenfrey line with the usual addition in R is a semitopological (in fact, paratopological) group which is not an S-topological group, because the inverse mapping i is not semi-continuous: is not semi-open. By [5, Example 2.6 (b)] the real line R with the usual addition and the co-finite topology is another such example (here the multiplication mapping is not semi-continuous). Example 5.1.22 in [10] (see [5]) is an S-topological group which is not a semitopological group. It is worth to mention that according to a result in [14] every paratopological S-topological group is a topological group. The inverse mapping i : G → G is continuous and hence semi-continuous. Therefore, (G, + 2 , τ ) is an S-topological group which is not a topological group. It was noticed in [3] that (G, + 2 , τ ) is not an s-topological group, and in [5] that it is not a semitopological group.
Remark 3.2. Let n > 2 be a natural number. Consider the cyclic group G = Z n = {0, 1, · · · , n − 1} of order n with the multiplication mapping m = + n -the addition modulo n. Take the topology τ = {∅, G, {0}} on G. Then (G, + n , τ ) is an S-topological group. Indeed, the preimage On the other hand, the inverse mapping i : G → G is continuous, and thus semi-continuous.     The following lemma will be used in the sequel. Also, we have the following (known) definition.
The following simple result is of fundamental importance in what follows. Proof. We prove the statement only for left translations. Of course, left translations are bijective mappings. We prove directly that for any x ∈ G the translation x is semi-continuous. Let y be an arbitrary element in G and let W be an open neighbourhood of x (y) = x * y = x * (y −1 ) −1 . By definition of s-topological groups there are semi-open sets U and V containing x and y −1 , respectively, such that U * V −1 ⊂ W . In particular, we have x * V −1 ⊂ W . By Lemma 3.1 the set V −1 is a semi-open neighbourhood of y, so that the last inclusion actually says that x is semi-continuous at y. Since y ∈ G was an arbitrary element in G, x is semi-continuous on G.
We prove now that   Proof. Take any elements x and y in G and put z = x −1 * y. Then z is an S-homeomorphism of G and z (x) = x * z = x * (x −1 * y) = y.
Hence f is irresolute (and thus semi-continuous) at the point x of G, hence on G, because x was an arbitrary element in G.
Theorem 3.6. Let (G, * , τ ) be an s-topological group with base β e at the identity element e such that for each U ∈ β e there is a symmetric semi-open neighbourhood V of e such that V * V ⊂ U . Then G satisfies the axiom of s-regularity at e.
Proof. Let U be an open set containing the identity e. Then, by assumption, there is a symmetric semi-open neighbourhood V of e satisfying V * V ⊂ U . We have to show that sCl(V ) ⊂ U . Let x ∈ sCl(V ). The set x * V is a semiopen neighbourhood of x, which implies x * V ∩ V = ∅. Therefore, there are points a, b ∈ V such that b = x * a, i.e. x = b * a −1 ∈ V * V −1 = V * V ⊂ U .