Quasi-boundedness of irresolute paratopological groups

Continuing the study of irresolute paratopological groups, our focus in this paper is to define and study the boundedness for irresolute paratopological groups. Premeager property for irresolute paratopological groups is discussed. It is proved that every open subgroup of a quasi-bounded, premeager irresolute paratopological group is premeager. For bounded homomorphisms on an irresolute paratopological group, new notions nbq -quasi bounded and bqbq-quasi bounded homomorphisms are introduced and discussed. Subjects: Advanced Mathematics; Foundations & Theorems; History & Philosophy of Mathematics


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Irresolute topological groups were defined and studied by Khan et al. in 2015. Further, in 2016 they introduced four classes of groups namely s-(S-, irresolute-, Irr-) paratopological groups. Each paratopologized group is defined in such a way that the topology is endowed upon a group such that the group operation satisfies certain condition which is either weaker or stronger than continuity. Azar defined the bounded topological group and established some relationships with other topological properties of the space. He proved that if a topological group G is metrizable then E⊆G is bounded with respect to topology if and only if it is bounded with respect to metric. Lin et al. in 2011 introduced pseudobounded and -pseudobounded paratopological groups and showed that when a paratopological group becomes a topological group. In present paper, we shall introduce the relevant concepts of boundedness for irresolute paratopological groups to be able to discuss and develop basic theory.
2008; Lin & Lin, 2011, Ravsky, 2001. A paratopological group G is a group G with a topology such that the product mapping of G × G into G is jointly continuous and a topological group G is a paratopological group such that the inverse mapping of G onto itself associating x −1 with arbitrary x ∈ G is continuous. It is well known that paratopological group is a good generalization of topological group.
Irresolute topological groups were defined and studied by Khan, Siab, and Ljubiša (2015). Further,  introduced four classes of paratopologized groups which are s-(S-, irresolute-, Irr-) paratopological groups. Each paratopologized group is defined in such a way that the topology is endowed upon a group (G, * ) such that the group operation satisfies certain condition which is either weaker or stronger than continuity. Semi-topological groups with respect to irresolute were defined and studied by Oner et al independently Oner, burc Kandemir, and Tanay (2013). Azar (2008) defined the bounded topological group G and established some relationships with other topological properties of G. He proved that if a topological group G is metrizable then E ⊆ G is bounded with respect to topology if and only if it is bounded with respect to metric. Lin & Lin (2011) introduced pseudobounded and -pseudobounded paratopological groups and showed that when a paratopological group becomes a topological group. In present paper, we shall introduce the relevant concepts of boundedness for irresolute paratopological groups in order to be able to discuss and develop basic theory.
Bounded homomorphisms and their algebraic and topological algebraic structure are of interest for their own right and also for their applications in other area of mathematics. Therefore, it will be of intereset to consider different types of bounded homomorphisms on irresolute paratopological groups. So, in this paper, we shall define the new notions nb q -quasi bounded and b q b q -quasi bounded of bounded homomorphisms on an irresolute paratopological groups.

Preliminaries
In 1963, Levine defined semi-open sets in topological spaces. Since then many mathematicians explored different concepts and generalized them using these sets (see Anderson & Jensen, 1967;Crossley & Hildebrand, 1971a;Nour, 1998;Piotrowski, 1979). A subset A of a topological space X is  (Crossley & Hildebrand, 1971a, 1971b. Let us mention that x ∈ sCl(A) if and only if for any semiopen set U containing x, U ∩ A ≠ �. Basic properties of semi-open sets are given in Levine (1963), and of semi-closed sets and the semiclosure in (Crossley & Hildebrand, 1971a, 1971b 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. A topological space (X, ), is said to be: semicompact (Carnahan, 1973;Gupta & Noiri, 2006;Sarak, 2009), if every semi-open cover of X has a finite subcover, locally semi-compact if and only if every point x has a semi-open neighbourhood U whose semi-closure is semi-compact or it is semi-compact in the small semi-open neighbourhoods.
A subset D of a topological space (G, ) is said to be dense if sCl(D) = X (Modak, 2011). In topological space (X, ), a set which cannot be expressed as the union of two semi-separated sets is said to be a semi-connected set. The topological space (X, ) is said to be semi-connected if and only if X is semi-connected.
Clearly, continuity implies semi-continuity; the converse need not be true. Notice that a mapping Then f is semi-continuous but not continuous.
Definition 2.2 A mapping f :X → Y between topological spaces X and Y is called: (2) quasi-open Lin and Lin (2011) (3) irresolute Crossley and Hildebrand (1971a) Definition 2.4 Let G be a paratopological group and E ⊆ G. We say that E is an pseudobounded Azar (2008) (resp. -pseudobounded Lin & Lin, 2011) subset of G, if for every neighbourhood U of the identity element e of G, there exists a natural number n such that E ⊆ U n (resp. E ⊆ ∪ n∈ℕ U n ). If G is an pseudobounded (resp. -pseudobounded) subset of G, then we say that G is pseudobounded (resp. -pseudobounded).
Definition 2.6 Lin and Lin (2011) Let G be a paratopological group. G is called premeager if, for any nowhere dense subset E of G, we have E n ≠ G for each n ∈ ℕ.

Quasi-bounded and -Quasi-bounded irresolute paratopological Groups
In this section, we will define quasi-bounded and -quasi-bounded irresolute paratopological groups by following Lin and Lin (2011). Note that "bounded" in Azar (2008) was called "pseudobounded" in Lin and Lin (2011) since bounded has another meaning as well in topological algebra. Proof Put F = G�{gU:g ∈ G, gU ∩ M = �}. Then, clearly, F is a semi-closed subset of G and M ⊂ F. Take any y ∈ F. Then yU ∩ M ≠ �, that is, yh=m, for some h ∈ U and m ∈ M. Hence, y = mh −1 ∈ MU −1 . Thus,

Theorem 3.3 Suppose that G is an irresolute paratopological group and not an irresolute topological group. Then there exists a semi-open neighbourhood U of the neutral element e of G such that U ∩ U −1
is semi-nowhere dense in G, that is, the semi-interior of the semi-closure of U ∩ U −1 is empty.
Proof By Theorem 3.19 (2) , the multiplication mapping m:G × G → G is an irresolute mapping, whereas the inverse operation in G is not irresolute. Therefore, by Theorem 3.25 , the inverse mapping is not irresolute at e, and we can choose a semi-open neighbourhood W of e such that e ∉ sint(W −1 ). Since the multiplication in G is irresolute, we can find a semi-open neighbourhood U of e such that U 3 ⊂ W. Claim. The set U ∩ U −1 is semi-nowhere dense in G. Assume the contrary. Then there exists a nonempty semi-open set V such that V ⊂ sCl(U ∩ U −1 ). Proof Let H be a quasi-bounded dense subgroup of an irresolute paratopological (G, ⋅, ). Take a semi-open neighbourhood U of the identity e in (G, ⋅, ). Since H is a quasi-bounded subset of (G, ⋅, ), there exists n ∈ ℕ such that H ⊆ U n , equivalently, H ⊆ U −n . Hence, using Theorem 3.2,

By Theorem 3.2, it follows that
there exists an n ∈ ℕ such that g ∈ V n , that is, the element g can be written in the form g = y 1 … y n , where y 1 , … , y n ∈ V. Since x commutes with each element of V by claim, we have g ⋅ x = y 1 … y n ⋅ x = y 1 … x ⋅ y n = … = y 1 ⋅ x … y n = x ⋅ y 1 … y n = x ⋅ g. Therefore, the element x ∈ H is in the centre of the group G. Because x is an arbitrary element in H, we conclude that the centre of G contains H. ✷

Theorem 3.6 Let G and H be irresolute paratopological groups and suppose that f: G → H is group isomorphism. If f is irresolute and E ⊆ G is quasi-bounded subset of G, then f(E) is quasi-bounded subset of H.
Proof Let V be a semi-open neighbourhood of e ∈ H. Then f −1 (V) is a semi-open neighbourhood of e. Since E is quasi-bounded subset of G, there is a natural number n, such that

Theorem 3.7 Let f :G → H be an irresolute homomorphism from an irresolute topological group G onto the irresolute topological group H. If G is quasi-bounded ( -quasi-bounded), then H is quasi-bounded ( -quasi-bounded).
Proof Suppose that (G, ⋅, ) is -quasi-bounded. Take  (U∕H) n = G∕H. We claim that ∞ ∪ n=1 U n = G. In fact, let x ∈ G. Case 1:x ∈ H. There exists an n ∈ ℕ such that x ∈ V n . Therefore, we have and hence there exists an m ∈ N such that xH ∈ (U∕H) m . Therefore, there exist points x 1 , … , x m ∈ U such that xH = x 1 … x m H. Hence, there exist an h ∈ H and an l ∈ ℕ such that xh ∈ U m and h ∈ V l . It follows that

Premeager irresolute paratopological groups
In this section, we will define and discuss the premeager property for irresolute paratopological groups.
Definition 4.1 Let (G, ⋅, ) be an irresolute paratopological group. G is called premeager if, for any its semi-nowhere dense subset A of G, we have A n ≠ G for each n ∈ ℕ.
The Sorgenfrey line X (X = ℝ) does not have the premeager property. In particular, the Euclidean line does not have the premeager property.
Proof Let C be the usual Cantor set in [0, 1]. It is well known that C is nowhere dense in X. By [Kharazishvili (2004) Proof Let A be any semi-nowhere dense subset of H. Suppose that there exists some n ∈ ℕ, such that A n = H. Therefore, (f −1 (A)) n = f −1 (A n ) = f −1 (H) = G. Since G is premeager, the set f −1 (A) is a nonsemi-nowhere dense subset of X. Hence there is a non-empty semi-open subset U of X such that U ⊂ sCl(f −1 (A)). It follows from definition (3)  Proof Let H be an open subgroup of G. Suppose that H is non-premeager. Then there exists a seminowhere dense subset A of H and an n ∈ ℕ such that A n = H. Since G is quasi-bounded, it follows that there is an m ∈ N such that H m = G. Hence (A n ) m = H m = G = A nm . However, the set A is a seminowhere dense subset of G, which is a contradiction. ✷

nb q -quasi bounded and b q b q -quasi bounded homomorphism
Bounded homomorphisms and their algebraic and topological algebraic structure are of interest for their own right and also for their applications in other area of mathematics. Therefore, it will be of intereset to consider different types of bounded homomorphisms on irresolute paratopological groups. So,in this section, we will define new notions nb q -quasi bounded and b q b q -quasi bounded of bounded homomorphisms on an irresolute paratopological group. (2) b q b q -quasi bounded if for every quasi bounded set B ⊂ G, (B) is quasi bounded in H.
The set of all nb q -quasi bounded (b q b q -quasi bounded) homomorphisms from an irresolute topological group G to an irresolute topological group H is denoted by Hom nb q (G, H)(Hom b q b q (G, H)). We write Hom(G) instead of Hom(G, G).