On p-topological groups

In this paper, we introduce the notions of p-topological group and p-irresolute topological group which are generalizations of the notion topological group. We discuss the properties of p-topological groups with illustrative examples and we provide a connected p-topological group on any group G whose cardinality is not equal to 2. Also, we prove that translations and inversion in p-topological group are phomeomorphism and demonstrate that every p-topological group is phomogenous which leads to check whether a topology on a group satisfies the conditions of p-topological group or not.


introduction
Topological group is a mathematical structure on a set which is defined by underlying two distinguished structures on that set namely group and a topology. While merging two distinguished structures, the way of approach is keeping one structure as fundamental and the other one as deciding factor. We come across so many types of mathematical structures binded together in this way such as ring, field, vector space, algebra, normed linear spaces, etc. Sophus Lie built up the vast theory of those topological groups which he called continuous groups and are also known as Lie groups [12]. He studied the case, for a group, how it is possible to define a topology such that the binary operations, group multiplication and inversion are continuous. Then he define continuous group, a group having continuous binary operation which the basic idea of topological groups and Lie groups emerged. A topological group in modern notion is defined as, a group together with a topology such that the binary operations -multiplication and inversion are continuous. Based on this, some generalizations of topological groups such as paratopological groups, semitopological groups and quasitopological groups are defined. In a finite group, all the above mentioned generalizations coincide [1].
The concepts of S-topological group and s-topological group were discussed in [10] and the theory of almost topological group was initiated in [11]. In this paper, we discuss some more generalizations which are defined based on pre-open sets and present a new generalization of topological group called p-topological group.

Preliminaries
Throughout this paper, the triple (G, ·, τ ) denotes a group (G, ·) together with a topology τ . For x, y ∈ G we write xy instead of x · y.
For any x ∈ G, x −1 denotes the inverse of x in G. Let A, B ⊆ G, then AB = {ab : a ∈ A, b ∈ B}. The notion of pre-open sets of a topology was introduced by Mashhour et. al in [2] and the notion of semi-open sets was defined by N. Levine in [14]. The notion of regular open sets was introduced by K. Kuratowski [8]. Let X be a topological space. For a set A, interior of A is denoted by int(A) and the closure of A is denoted by cl(A). A subset A of X is said to be pre-open [2] (respectively, semi-open [14] and regular open [11]) if A ⊆ int(cl(A)) (respectively, A ⊆ cl(int(A)) and A = int(cl(A))). The largest pre-open set contained in A is termed as pre-interior of A [17] and the smallest pre-closed set containing A is called as pre-closure of A [17]. Pre-interior and pre-closure of A are denoted by pint(A) and pcl(A). For a set S, the power set of S is denoted by P(S) and for a topology τ , the collection of pre-open sets is denoted by τ p .
Definition 2. Let G be a group and τ be a topology on G. Then the pair (G, τ ) is said to be topological group [3] (respectively, s-topological group [10], almost topological group [11]) if multiplication and inversion are continuous (respectively, semi-continuous, almost continuous). The pair (G, τ ) is said to be paratopological group [3] (respectively, semitopological group) if multiplication is continuous (respectively, semi-continuous). Let x, y ∈ G then (G, τ ) is called a s-topological group [10] if for each open neighbourhood U of xy −1 there exist semi-open neighbourhoods V of x and W of y such that V W −1 ⊆ U .

Lemma 1 ([7]
). Let (X i ) i∈I be a family of topological spaces and ∅ = A i ⊆ X i for each i ∈ I. Then, i∈I A i is pre-open in i∈I X i if and only if A i is pre-open in X i for each i ∈ I and A i is non-dense for only finitely many i ∈ I. Lemma 2 ( [5]). Let X be a topological space and A ⊆ X. Then:

p-topological groups and their basic properties
In this section, we introduce the concept of p-topological group and investigate its basic properties with illustrative examples. We observe that -Any group with partition topology is a trivial example of p2-topological group. -Every finite left (right) topological group is p-topological group.
Since the basis of topology on finite left (right) topological group is cosets of a subgroup (partition topology) [1] and for the partition topology τ on any set X, τ p is P(X ) and so every subset of X is pre-open. -Every topological group is p-topological group, but converse need not be true. The following Examples 3.2, 3.4, 3.5 are all a p-topological group but not a topological group. One may ask the question that, if there is any topology τ for a group G whose τ p = P(X ) such that satisfies the conditions of p-topological group. An example is given below which answers this. Multiplication: Inverse: Thus (G, ⊕, τ ) is a p-topological group. Let us change the topology on G and check whether p-topological group or not. A finite group with indiscrete topology is the only connected finite topological group. But in the case of p-topological group we can provide some more connected topologies. For example, the above mentioned p-topological group is connected. We can define a topology τ on G, for any group G such that (G, τ ) is connected p-topological group as follows.
As in the above example, we can provide more connected p-topological groups for an infinite group as follows :  Thus, multiplication is pre-continuous on G. Now, the pre-continuity of inversion follows: Hence (G, τ ) is a connected p-topological group. By the above example, We can see that, in an infinite group G, for any n ∈ N we can provide n number of connected topologies τ i , i = 1, 2, 3, . . . , n such that (G, τ i ) is p-topological group.
The proof is trivial by the facts that, A is open ⇔ intA = A and pcl(A −1 ) ⊆ cl(A −1 ) (Lemma 2.4 (ii)). Though the result is trivial an interesting fact is as follows.
In the above proposition, the set A cannot be assumed to be pre-open and the reverse containment need not be hold.
Similarly we can prove that Ax is pre-open.
In the above result, the openness of A cannot be extended to pre-openess. Indeed, consider (G, τ ) in Example 3.2 which is a p-topological group. Let {1} ∈ τ p and 2 ∈ G.
implies the nonempty set A ∩ W . Thus c is a limit point of A. Since A is closed we have c ∈ A. Now b = xc and so b ∈ xA. By the above argument, pcl(xA) ⊆ xA and since xA ⊆ pcl(xA) is trivial we have xA = pcl(xA). Hence xA is pre-closed. Proof of Ax is similar. Theorem 1. Let A be any subset of a p-topological group G. Then: (iv) Let b ∈ int(xA). Then b = xa for some a ∈ A. We know that multiplication is pre-continuous. Then there exist pre-open neighbourhoods U and V of x and a such that ( there exist c ∈ A ∩ U and d ∈ B ∩ V . Now cd ∈ (AB) ∩ (U V ) ⊆ AB ∩ W which implies that AB ∩ W = ∅. Hence x is a limit point of AB and so x ∈ cl(AB).
Hence inversion is p-homeomorphism. Definition 5. A topological space X is said to be a p-homogeneous space if for any x, y ∈ X, there exists a p-homeomorphism f such that f (x) = y.
Proof. Let a, b ∈ G and c = ba −1 . By Theorem 3.14, each translation in p-topological group is p-homeomorphism. Thus we have λ c (a) = ca = ba −1 a = b. Hence, G is p-homogeneous.
The reason behind Theorem 3.16 is that, it is harder to decide, whether a topology on a group G satisfies the required conditions of p-topological group or not by checking pre-continuity on each elements. In a homogeneous space, all points behave in the same way. This observation suggests that, at first we have to define a basis at the identity element e. Then move the basis by means of translations to obtain a pre-open base at each element of the group G.
Theorem 6. Let G be a p-topological group and H be a subgroup of G. One may note that, by Lemma 2.3, For a family of topological spaces Proof. (i) Let U ∈ U. Then U is an open neighbourhood of e. We know that e = e.e. Since G is a p-topological group, the mapping (x, y) → xy is pre-continuous and so there exists pre-neighbourhoods P and Q of e such that P Q is contained in U . Let V be the smallest pre-neighbourhood among P and Q and so there exists V ∈ U p such that V 2 ⊂ U .
(ii) Let U ∈ U. Then U is an open neighbourhood of e. We know that the inverse of e is itself. Since G is a p-topological group, the mapping x → x −1 is pre-continuous and so there exists a pre-neighbourhood V of e such that V −1 ⊂ U .
(iii) Let U ∈ U and x ∈ U . We know that x = x.e (x = ex). Since G is a p-topological group, the mapping (x, y) → xy is pre-continuous and so there exist pre-neighbourhoods P of x and Q of e such that P Q(QP ) is contained in U . So for all x ∈ U , there is a V ∈ U p such that xV ⊂ U (V x ⊂ U ).
(iv) Let U ∈ U and x ∈ G. We know that xex −1 = e. Since G is a p-topological group, each translation is a p-homeomorphism of G and so the map e → xex −1 is a p-homomorphism of G. Hence for every U ∈ U and x ∈ G, there is a V ∈ U p such that xV x −1 is contained in U . ( Thus pcl(V ) ⊆ U . By Lemma 2.6 (ii), G satisfies p-regularity at e.

p-irresolute topological groups and pre-connectedness
In this section, we discuss the independency of p-topological group from other generalization concepts of topological group. We also explore preconnectedness properties of p-irresolute topological group.  Example 8. Consider (G, τ ) in Example 3.2, which is a p-topological group.
-Since the only regular open set in G is itself, we have (G, τ ) is an almost topological group. -Since every s-topological group is S-topological we have (G, τ ) is not a s-topological group.
Definition 6. The pair (G, τ ) is said to be p-irresolute topological group if multiplication and inversion mappings are pre-irresolute.
A topological space X is said to be pre-connected [16] if X cannot be written as the union of two disjoint nonempty pre-open sets.
Example 10. Consider R with usual topology which is connected. Here Q and Q c are disjoint pre-open sets [Q ⊂ int(cl(Q)), Q c ⊂ int(cl(Q c ))] whose union is R. Hence R with usual topology is not pre-connected.
Theorem 10. Let G be a p-irresolute topological group and H be a subgroup of G. If H, G/H are pre-connected, then G is pre-connected.
Proof. Let us assume that G = U ∪ V where U and V are disjoint nonempty pre-open sets. Since H is pre-connected, each coset of H is either a subset of U or a subset of V . Thus, the relation It expresses G/H as the union of disjoint nonempty pre-open sets which is a contradiction to pre-connectedness of G/H. Thus, G is pre-connected.
Theorem 11. Let G be a pre-connected p-irresolute topological group and e be its identity element. If V is any pre-open neighbourhood of e, then G is generated by V .
Proof. Let V be a pre-open neighbourhood of e. For each n ∈ N, we denote V n by the set of elements of the form v 1 · v 2 · · · v n where each v i ∈ V . Let U = ∪ ∞ n=1 V n . Since G is pre-connected, suppose if we prove U is pre-open and pre-closed, we have G = U and so G is generated by V . Since each V n is pre-open and arbitrary union of pre-open sets is pre-open, U is pre-open. Let us prove that U is pre-closed. Let x ∈ pcl(U ). Since xV −1 is a pre-open neighbourhood of x, it must intersect U . Thus, let y ∈ U ∩ xV −1 . Since y ∈ xV −1 then y = xv −1 for some v ∈ V . Since y ∈ U then y ∈ V n for some n ∈ N which implies y = v 1 v 2 · · · v n with each v i ∈ V . Now, we have x = v 1 v 2 · · · v n v and so x ∈ V n+1 ⊆ U . Hence U is pre-closed. Since G is pre-connected and U is pre-open and pre-closed we have U = G. Thus, G is generated by V .
Theorem 12. If G is a pre-connected, p-irresolute topological group and N , a discrete invariant subgroup of G, then N ⊆ Z(G), where Z(G) denotes the center of the group G.
Proof. Suppose the invariant subgroup N = {e}, then the result is trivial. Suppose that the subgroup N is non-trivial. Let x ∈ N be an arbitrary element of G distinct from the identity element e. Since the group N is discrete, we can find an open neighbourhood U of x in G such that U ∩ N = {x}. Since every open set is pre-open and by definition of p-irresolute topological group, there exist a pre-open neighbourhood V of e and a preopen neighbourhood V x of x in G such that (V x)V −1 ⊂ U . Let a ∈ V be arbitrary. Since N is an invariant subgroup of G, we have aN = N a which implies that ax ∈ N a and so axa −1 ∈ N . It is also clear that axa −1 ∈ V xV −1 ⊂ U . Therefore, axa −1 ∈ U ∩ N = {x} which implies axa −1 = x.
Thus, ax = xa for each a ∈ V . Since the group G is pre-connected, V n with n ∈ N covers the group G. Thus, every element b ∈ G can be written in the form b = a 1 a 2 · · · a n where a 1 , a 2 , . . . , a n ∈ V and n ∈ N. Since x commutes with every element of V , we have bx = a 1 a 2 · · · a n x = a 1 a 2 · · · xa n . . .
Hence the element x ∈ N is in the center of group G. Since x is an arbitrary element of N , we proved that the center of G contains N .

Conclusion
Topological groups mostly deals with an infinite set alone by an assumption in separation. To overcome this, some generalizations of topological groups are defined but they did not attain similar properties to topological group. By defining p-topological group, we reach a space which has properties close relevant to topological group.