On almost topological groups

We introduce and study the almost topological groups which are fundamentally a generalization of topological groups. Almost topological groups are defined by using almost continuous mappings in the sense of Singal and Singal. We investigate some permanence properties of almost topological groups. It is proved that translation of a regularly open (resp. regularly closed) set in an almost topological group is regularly open (resp. regularly closed). And this fact gives us a lot of important and useful results of almost topological groups.


Introduction
A topological group is a group endowed with a topology which turns out the group operation and the inversion mapping (that is, x → x −1 ) continuous. Since the advent of this concept, it has been captured a great attention from different researchers and mathematicians. Many mathematicians have contributed significantly in the field of topological groups. Some Recent literature of mathematics contains several similar notions and generalizations of topological groups. Semi-topological groups [10,11], stopological groups [1,4,8], S-topological groups [1], quasi S-topological groups [6], irresolute topological groups [5,9], quasi-irresolute topological groups [12] and paratopological groups [16] are well-known.
In this paper, we will study a new class of spaces, which we shall call almost topological groups. Basically, almost topological groups are a generalization of topological groups. We will present some examples of almost topological groups which are not topological groups. We also set forth some basic properties of almost topological groups. It is proved that translation of a regularly open (resp. regularly closed) set in an almost topological group is regularly open (resp. regularly closed), that any group homomorphism of almost topological groups which is R-continuous at the identity is almost continuous everywhere.

Preliminaries
Throughout the present paper, (X, τ ) (or simply X) means a topological space. For A ⊆ X, Cl(A) denotes the closure of A and Int(A) denotes the interior of A. By f : X → Y , we denote a map f from a topological space X to a topological space Y .
In A subset A of a topological space X is said to be δ−open [17] if for each x ∈ A, there exists a regular open set U in X such that x ∈ U ⊆ A. The complement of a δ−open set is called δ−closed set [17]. The intersection of all δ−closed sets in X containing a subset A ⊆ X is called the δ−closure of A and is denoted by Cl δ (A). It is known that a subset A of X is δ−closed if and only if A = Cl δ (A). A point x ∈ Cl δ (A) if and only if A ∩ Int(Cl(U )) = ∅ for each open set U in X containing x. The union of all δ−open sets in X that are contained in A ⊆ X is called the δ−interior of A and is denoted by Int δ (A). The family of all regularly open (resp. regularly closed) sets of X is denoted by RO(X) (resp. RC(X)). If A ∈ RO(X) and B ∈ RO(Y ), then A × B ∈ RO(X × Y ), where X, Y are topological spaces.
In 1968, Singal and Singal [15] defined and investigated the notion of almost continuous mappings. They defined a function f : X → Y is said to be almost continuous if, for each x ∈ X and for each open neighborhood V of f (x) in Y , there exists an open neighborhood U of x such that f (U ) ⊆ Int(Cl(V )). In the same work, they proved a classical result saying that a function f : X → Y is almost continuous if and only if f −1 (V ) is open (resp. closed) in X, for every regular open (resp. regular closed) set V in Y . Since then, many authors have accomplished an excellent work in the field of almost continuous mappings.

Almost topological groups
Before we start our main work, we introduce some notations. By G, we mean the group (G, * ) where ' * ' is a binary operation on G under which G is a group. For x, y ∈ G, we denote x * y by xy unless stated explicitly. We denote the inverse of an element x in G by x −1 (i.e., under addition operation on G, x −1 means −x and under multiplication operation, x −1 retains its notation and meaning). Translation of a set A in a group (G, * ) (or simply, G) by an element g is denoted by gA, defined by gA = {g * a : a ∈ A}. In this section, we introduce the notion of almost topological groups, present hosts of examples of almost topological groups and investigate several of their basic properties. Along with other results, we prove that translation of a regularly open (resp. regularly closed) set in an almost topological group is regularly open (resp. regularly closed).
we denote A * B by gB and B * A by Ag.
Before we start discussing some general properties of almost topological groups. Let us present some examples of them.  In general, it is obvious from the definition that every topological group is an almost topological group, but the converse is not true in general. The following are the examples of almost topological groups which are not topological groups.
Then (G, τ ) is an almost topological group which is not a topological group.
In fact, if G is any group, A proper subset of G and τ = {∅, A, G}. Then (G, τ ) is an almost topological group which is not a topological group. Henceforth, we denote an almost topological group by (G, τ ) (or simply G when there is no chance of confusion). We turn now to some general properties of almost topological groups.
Theorem 3.1. Let (G, τ ) be an almost topological group and let g ∈ G be any element of G. Then: (1) the mapping h g : G → G defined by h g (x) = gx, ∀x ∈ G, is almost continuous; (2) the mapping l g : G → G defined by l g (x) = xg, ∀x ∈ G, is almost continuous. Proof.
(1) Let x be any element of G and let W be a regular open set in G containing gx. By definition 3.1, there exist open neighborhoods U and V of g and x respectively, in G such that U V ⊆ W . In particular, gV ⊆ W which means that h g (V ) ⊆ W . This indicates that h g is almost continuous at x and hence h g is almost continuous.
(2) Pick up x ∈ G and let W ∈ RO(G) containing xg.
Then there exist open sets U in G containing x and V in G containing g such that U V ⊆ W . This gives U g ⊆ W , i.e., l g (U ) ⊆ W . This shows that l g is almost continuous at x. Since x was an arbitrary element of G, l g is almost continuous. (1) gA ∈ RO(G), for all g ∈ G.
(1) Firstly, we show that gA ∈ τ . Let x ∈ gA be any element. Then by definition of almost topological groups, there exist open neighborhoods U of g −1 and V of x in G such that U V ⊆ A. In particular, g −1 V ⊆ A. This is equivalent to the relation V ⊆ gA. This indicates that x ∈ Int(gA) and thus, Int(gA) = gA. That is, gA ∈ τ . Consequently, gA ⊆ Int(Cl(gA)). With an eye to the necessity of the problem, we have to show that Int(Cl(gA)) ⊆ gA.
(2) The proof follows by similar arguments as in part (1) Corollary 3.1. Let F be any regularly closed set in an almost topological group G. Then: (1) gF ∈ RC(G), for each g ∈ G.
Proof. We only prove (1)  Using similar arguments, we obtain the following result: Theorem 3.7. Under the same hypothesis of Theorem 3.6, the following are valid: (1) Int(gA) = gInt(A), for each g ∈ G.
(2) Int(Ag) = Int(A)g, for each g ∈ G. Analogously, (2) and (3) can be proved. In a similar style, we obtain the following result.
Theorem 3.9. Let B be any closed subset of an almost topological group. Then Cl(gCl(Int(B))) ⊆ gB for each g ∈ G.
Theorem 3.10. Let G be an almost topological group. For any A ⊆ G, the following hold: (1) Cl(gA) ⊆ gCl δ (A), for each g ∈ G.
(1) Let x be ny element of Cl(gA). Consider y = g −1 x and let W be any open neighborhood of y in G.
Then there exist open neighborhoods U of g −1 and V of x in G such that U V ⊆ Int(Cl(W )). Since x ∈ Cl(gA), there is p ∈ (gA) ∩ V and thus we always have, g −1 p ∈ A∩(U V ) ⊆ A∩Int(Cl(W )) ⇒ A∩Int(Cl(W )) = ∅. Therefore, y ∈ Cl δ (A); that is, x ∈ gCl δ (A).
(2) Let y ∈ gInt δ (A). Then y = gx for some x ∈ Int δ (A). This means that there exists a regular open set U in G such that x ∈ U ⊆ A. In a sense, Int(Cl(U )) ⊆ A ⇒ gInt(Cl(U )) ⊆ gA. This gives y = gx ∈ gInt(Cl(U )) ⊆ gA. By Theorem 3.2, gInt(Cl(U )) is open and thus, y ∈ Int(gA). Hence gInt δ (A) ⊆ Int(gA).      Proof. Let x ∈ Cl(U ). Since xU is open neighborhood of x, U ∩ (xU ) = ∅. Therefore, we obtain an equation, g = xh for some g, h ∈ U . This gives x = h −1 g ∈ U −1 U . Hence the assertion follows.
Definition 3.4. A topological space X is called almost regular [13] if for each regularly closed set F in X and each x / ∈ F , there exist disjoint open sets U and V in X such that x ∈ U and F ⊆ V .
Conjecture. Every almost topological group is almost regular. Definition 3.5. A topological space X is called nearly compact [14] if every open cover of X has a finite subcover the interiors of the closures whose members cover X. Theorem 3.16. If A and B are compact subsets of an almost topological group G, then AB is nearly compact.
Proof. Let A and B be compact subsets of G. Since the image of a compact set under almost continuous mappings is nearly compact, by definition of an almost topological group, it follows that AB is nearly compact.

Acknowledgement
I would like to thank the anonymous reviewers for their valuable comments/suggestions regarding the improvement of this manuscript. I must thank the editor for his kind cooperation throughout process of this manuscript.