Abstract
In this paper, we continue the study of s-topological and irresolute-topological groups. We define semi-quotient mappings which are stronger than semi-continuous mappings, and then consider semi-quotient spaces and groups. It is proved that for some classes of irresolute-topological groups (G, *, τ) the semi-quotient space G/H is regular. Semi-isomorphisms of s-topological groups are also discussed.
1 Introduction
The basic aim of this article is to study properties of topological spaces and mappings between them by weakening the continuity and openness conditions. Semi-continuity [1] and irresolute mappings [2] were a consequence of the study of semi-open sets in topological spaces. In [3] Bohn and Lee defined and investigated the notion of s-topological groups and in [4] Siddique et. al. defined the notion of 5-topological groups. In [5] Siab et. al. defined and studied the notion of irresolute-topological groups by using irresolute mappings. Study of s-paratopological groups and irresolute-paratopological groups is a consequence of the study of paratopological groups (see [6]). For the study of semi-topological groups with respect to semi-continuity and irresoluteness we refer the reader to Oner’s papers [7-9].
In this paper we continue the study of properties of s-topological and irresolute-topological groups. Keeping in mind the existing concepts, semi-quotient topology on a set is defined as a generalization of the quotient topology for spaces and groups. Various results on semi-quotients of topologized groups are proved. A counter example is given to show that the quotient topology is properly contained in the semi-quotient structure. We define also semi-isomorphisms and S-isomorpisms between topologized groups and prove that if certain irresolute-topological groups G and H are semi-isomorphic or S-isomorphic, then their semi-quotients are semi-isomorphic. Investigation of s-openness and s-closedness of mappings on s-topological groups is also presented.
2 Definitions and preliminaries
Throughout this paper X and Y are always topological spaces on which no separation axioms are assumed. If ƒ : X → Y is a mapping between topological spaces X and Y and B is a subset of F, then f←(B) denotes the pre-image of B. By C1(A) and Int(A) we denote the closure and interior of a set A in a space X. Our other topological notation and terminology are standard as in [10]. If (G, *) is a group, then e or eG denotes its identity element, and for a given x ∈ G, ℓx : G → G, y ↦ x ∘y, and rx : G→G, y ↦ y ∘ x, denote the left and the right translation by x, respectively. The operation * we call the multiplication mapping m : G x G → G, and the inverse operation x ↦ x–1 is denoted by i.
In 1963, N. Levine [1] defined semi-open sets in topological spaces. Since then, many mathematicians have explored different concepts and generalized them by using semi-open sets (see [2,11-14]). A subset A of a topological space X is said to be semi-open if there exists an open set U in X such that U ⊂ A ⊂ Cl(U), or equivalently if A ⊂ Cl(Int(A)). SO(X) denotes the collection of all semi-open sets in X, and SO(X, x) is the collection of semi-open sets in X containing the point x ∈X. The complement of a semi-open set is said to be semi-closed; the semi-closure of A ⊂ X, denoted by sCl(A), is the intersection of all semi-closed subsets of X containing A [15, 16]. x ∈ sCl(A) if and only if any U ∈SO(X, x) meets A.
Clearly, every open (resp. closed) set is semi-open (resp. semi-closed). It is known that a union of any collection of semi-open sets is again a semi-open set. The intersection of two semi-open sets need not be semi-open whereas the intersection of an open set and a semi-open set is semi-open. Basic properties of semi-open sets and semi-closed sets are given in [1], and [15, 16].
Recall that a set U ⊂X is a semi-neighbourhood of a point x ∈X if there exists A ∈SO(X, x) such that A ⊂U. 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 set A ⊂ X is semi-open in X if and only if A is a semi-open neighbourhood of each of its points. Let X be a topological space and A ⊂X. Then x∈ X is called a semi-interiorpoint of A if there exists a semi-open set U such that x ∈U ⊂ A The set of all semi-interior points of A is called a semi-interior of A and is denoted by sInt(A). A nonempty set A is pre-open (or locally dense) [17] if A ⊂ Int.(Cl(A)). A space X is s-compact [18], if every semi-open cover of X has a finite subcover. Every s-compact space is compact but converse is not always true. For some applications of semi-open sets see [13].
A mapping f : X → Y between topological spaces X and Y is called:
- semi-continuous [1] (resp. irresolute [2]) if for each open (resp. semi-open) set V ⊂Y the set f←(V). is semi-open in X. Equivalently, the mapping f is semi-continuous (irresolute) if for each x ∈X and for each open (semi-open) neighbourhood V of f(x), there exists a semi-open neighbourhood U of x such that f(U) ⊂V;
- pre-semi-open [2] if for every semi-open set A of X, the set f(A) is semi-open in Y;
- s-open (s-closed) if for every semi-open (semi-closed) set A of X, the set f(A) is open (closed) in Y;
- s-perfect if it is semi-continuous, s-closed, surjective, and f←(y) is s-compact relative to X, for each y in Y.
- S-homeomorphism [4] if f is bijective, semi-continuous and pre-semi-open.
We need also some basic information on (topological) groups; for more details see the excellent monograph [20]. If G is a group and H its normal subgroup, then the canonical projection of G onto the quotient group G/H (sending each g ∈G to the coset in G/H containing g) will be denoted by p. A mapping f : G → H between two topological groups is called a topological isomorphism if f is an algebraic isomorphism and a topological homeomorphism.
Let f : X → Y be a surjection; a subset C of X is called saturated with respect to f (or f-saturated) if f←(f(C) = C [21].
3 Semi-quotient mappings
A mapping f : X →Y from a spaceXonto a spaceYis said to be semi-quotient provided a subset V of Y is open in Y if and only if f ← (V) is semi-open in X.
Evidently, every semi-quotient mapping is semi-continuous and every quotient mapping is semi-quotient. The following simple examples show that semi-quotient mappings are different from semi-continuous mappings and quotient mappings.
Let X = Y = {1, 2, 3} and let τX = {∅, X, {1}, {2} {1, 2}, {1, 3}} and τY = {∅,X, {1}, {2}, {1 2}} betopologies on X and Y. Let f : X → Y be defined by f(x) = x, x ∈X. Since τY ⊂τχ, the mappingƒ iscontinuous, hence semi-continuous. On the other hand, this mapping is not semi-quotient because f← ({3}) is semi-open in X although {1, 3} is not open in Y.
Let X = {1, 2, 3, 4}, Y = {a, b}, τχ = {∅, X, {1}, {3}, {1, 3}}, τγ = {∅, Y, {a}}. Define f : X → Y by; f (1) = f(3) = f (4) = a; f (2) = b. The mapping fis not aquotient mapping because it is not continuous. On the other hand, f is semi-quotient: the only proper subset of Y whose preimage is semi-open in X is the set fag which is open in Y.
The following proposition is obvious.
(a) Every surjective semi-continuous mapping f : X → Y which is either s-open or s-closed is a semi-quotient mapping.
(b) If f : X → Y is a semi-quotient mapping and g : Y → Za quotient mapping, then g ∘ f : X → Zis semi-quotient.
We prove only (b). A subset V ⊂Z is open in Z if and only if g←(V) is open in Y (because g is a quotient mapping), while the latter set is open in Y if and only if f(←g←(V)) is semi-open in X (because f is semi-quotient). So, V is open in Z if and only (g ∘ f)←(V) is semi-open in X, i.e. g ∘ f is a semi-quotient mapping.
The restriction of a semi-quotient mapping to a subspace is not necessarily semi-quotient. Let X and Y be the spaces from Example 3.3, and A = {2, 4}. Then τA = {∅, A}. The restriction fA : A → Y of f to A is not a semi-quotient mapping because fA←{(a)} = {4} is not semi-open in A.
To see when the restriction of a semi-quotient mapping is also semi-quotient we will need the following simple but useful lemmas.
([22, Theorem 1]). Let X be a topological space, X0 ∈ SO(X) and A ⊂X0. Then A ∈ SO(X0 ) if and only if A ∈ SO(X).
([23, Lemma 2.1]). Let X be a topological space, X0 a subspace of X. If A ∈ SO(X0), then A = B ⋂ X0, for some B ∈ SO(X).
([21]). Let f : X → Y bea mapping, A a subspace of X saturated with respect to f, B a subset of X. If g : A → f(A) is the restriction of f to A, then:
(1) g←(C) = f←(C) for any C ⊂f(A);
(2) f(A⋂B) = f(A) ⋂f(B).
Now we have this result.
Let f : X → Y be a semi-quotient mapping and let A be a subspace of X saturated with respect to f, and let g : A →f(A) be the restriction of f to A. Then:
(a) If A is open in X, then g is a semi-quotient mapping;
(b) Iff is an s-open mapping, then g is semi-quotient.
(a) Let V be an open subset of f(A). Then V = W ⋂f(A) for some open subset W of X, so that g←(V) = f← (W⋂f(A)) = f←(W) ⋂A is a semi-open set in A.
Let now V be a subset of f(A) such that g←(V) is semi-open in A. We have to prove that V is open in f(A). Since g(V) is semi-open in A and A is open in X we have that g←(V) is semi-open in X. By Lemma 3.7, g←(V)= f←(V); the set f←(V) is semi-open in X since f is semi-quotient, hence g←(V) is semi-open in f(A). This means that V is open in Y and thus in f(A). This completes the proof that g is a semi-quotient mapping.
(b) Let now f be s-open and V a subset of f(A) such that g←(V) is semi-open in A. Again we must prove that V is open in f(A). Since g←(V)= f←(V) and g←(V) is semi-open in A, by Lemma 3.6 we have f←(V) = U ∩ A, for some U semi-open in X. As f is surjective, it holds f(f←(V) = V. By Lemma 3.7, then V = f(f←(V)) = f(U ∩ A) = f(U) ∩ f(A). The set f(U) is open in Y because f is s-open, so that V is open in f(A). Other part is the same as in (a).
As a complement to Proposition 3.4 we have the following two theorems.
Let X, Y and Z be topological spaces, f : X → Y a semi-quotient mapping, g : Y → Z a mapping. Then the mapping g o f : X → Zis semi-quotient if and only if g is a quotient mapping.
If g is a quotient mapping, then g ∘ f is semi-quotient as the composition of a semi-quotient and a quotient mapping (Proposition 3.4).
Conversely, let g o f be semi-quotient. We have to prove that a subset V of Z is open in Z if and only ifg←(V) is open in Y. For, the set (g ∘f)←(V) is semi-open in X and since f is a semi-quotient mapping we conclude that it will be if and only if g←(V) is open in Y.
Let f : X → Ybe a mapping and g : X → Z a mapping which is constant on each set f←({y}), y ∈Y. Then g induces a mapping h : Y → Z such that g = h ∘ f. Then:
(1) If f is pre-semi-open and irresolute, then h is a semi-continuous mapping if and only if g is semi-continuous;
(2) If f is semi-quotient, then h is continuous if and only if g is semi-continuous.
Since g is constant on the set f←({y}), y ∈ Y, then for each y ∈ Y, the set g(f←({y})) is a one-point set in Z, say h(y). Define now h : Y→Z by the rule
h(y) = g(f←(y)) (y ∈Y).
Then for each x ∈ X we have
g(x) = g(f←(f(x))) = h(f(x)), i.e. g = h º f.
(1) Suppose g is a semi-continuous mapping. If V is an open set in Z, then h←(V) = f(g←(V)) ∈ SO(Y) because f is pre-semi-open and g←(V) is semi-open in X. Thus h is a semi-continuous mapping.
Conversely, suppose h is a semi-continuous. Let V be an open set in Z. The set g←(V) = f←h←(V)) is semi-open in X because f is irresolute and h←(V) ∈SO(Y). So, g is semi-continuous.
(2) If g is semi-continuous, then for any open set V in Z we have g←(V) is a semi-open set in X. But, g←(V) = f←(h←(V)). Since f is semi-quotient it follows that h←(V) is open in Y. So, h is continuous.
Conversely, suppose h is continuous. For a given open set V in Z, h←(V) is an open set in Y. We have then g←(V) = f←(h←(V)) is semi-open in X because f is semi-quotient. Hence g is semi-continuous.
At the end of this section we describe now a typical construction which shows how the notion of semi-quotient mappings may be used to get a topology or a topology-like structure on a set.
Construction: Let X be a topological space and Y a set. Let f : X→ Y be a mapping. Define
sτQ := {V ⊂Y : f←(V) ∈ SO(X)},
It is easy to see that the family sτQ ia a generalized topology on Y (i.e. ∅ ∈ sτQ and union of any collection of sets in sτQ is again in sτQ) generated by f; we call it the semi-quotient generalized topology. But sτQ need not be a topology on Y. It happens if X is an extremally disconnected space, because in this case the intersection of two semi-open sets in X is semi-open [24]. It is trivial fact that in the latter case sτQ is the finest topology ς on Y such that f : X → (Y, σ) is semi-continuous. In fact, f : X → (Y, sτQ) is a quotient mapping in this case.
In particular, let p be an equivalence relation on X. Let p : X →X/ρ be the natural (or canonical) projection from X onto the quotient set X/ρ: for each x in X, p sends x to the equivalence class ρ(x). Then the family sτQ generated by p is a generalized topology on the quotient set Y/ρ, and a topology when X is extremally disconnected. This topology will be called the semi-quotient topology on X/ρ. Observe, that we forced the mapping p to be semi-continuous, that is semi-quotient.
This kind of construction will be applied here to topologized groups: to s-topological groups and irresolute-topological groups.
The following example shows that a quotient topology on a set generated by a mapping and the semi-quotient (generalized) topology generated by the same mapping are different.
Let the set X = {1, 2, 3, 4} be endowed with the topology
τ = (∅,X, {1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}.
Then the set SO(X) is
(∅, X, {1},{2},{3},{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4},{2,3,4}}.
Define the relation R on X by xRy if and only if x + y is even. Therefore,
R = {(1,1),(1,3),(2,2),(2,4),(3,1),(3,3),(4,2),(4,4),}
is an equivalence relation, and X/R = {R(1),R(2)} = {{1,3},{2,4}}. Let p : X →X/R be the canonical projection. Then, p←(R(1)) = {1, 3} ∈ SO(X), and p←(R(2)) = f2; 4} ∈ SO(X) , so that
sτQ = {∅, X/R, {R(1)},{R(2)}}
is the semi-quotient topology on X/R. On the other hand, p←({R(1)}) = {1,3} ∈ τ, but p←({R(2)}) = {2,4} ∉ τ. Therefore, the quotient topology on X/R is
τQ = {∅ X/R, {R(1)}}.
4 Topologized groups
In this section we give some information on s-topological groups and irresolute-topological groups introduced and studied first in [4] and [5], respectively.
([3]). An s-topological group is a group (G, *) with a topology τ such that for each x, y ∈G and each neighbourhood W of x * y—1there are semi-open neighbourhoods U of x and V of y such that U * V—1 ⊂W.
([5]). A triple (G, *, τ) is an irresolute-topological group if (G, *) is a group, and τ a topology on G such that for each x, y ∈G and each semi-open neighbourhood W of x * y—1 there are semi-open neighbourhoods U of x and V of y such that U * V—1 ⊂W.
([25]). If (G, *, x) is an s-topological group, y ∈G, and K an s-compact subset of G, then y * K–1is s-compactin G. Inparticular, K–1is s-compact.
If a mapping f : X →Y between topological spaces X and Y is s-perfect, then for any compact subset K of Y, the pre-image f←(K) is an s-compact subset of X.
Let {Ui : i ∈ } be a semi-open cover of f←(K). Then for each x ∈K the set f←(x) can be covered by finitely many Ui; let U(x) denote their union. Then O(x) = Y \ f [X \ ⊂ U(x)] is an open neighbourhood of x in Y because f is an s-closed map. So,
The following results are related to s-topological groups, and they are generalizations of some results for topological groups.
Let G, H and K be s-topological groups, φ : G → H a semi-continuous homomorphism, ψ G →K an irresolute endomorphisms, such that ker ψ ⊂ ker φ. Assume also that for each open neighbourhood U of eH there is a semi-open neighbourhood V of eK with ψ←(V) ⊂ ψ←(U). Then there is a semi-continuous homomorphism f : K → H such that φ = f ∘ ψ.
The existence of a homomorphism f such that φ = f ∘ ψ is well-known fact in group theory. We verify the semi-continuity of f. Suppose U is an open neighbourhood of eH in H. By our assumption, there is a semi-open neighbourhood V of eK in K such that ψ←(V) ⊂ φ←(U). Then φ = f ∘ ψ implies f(V) = φ(ψ←(V)) ⊂ φ(φ←U) ⊂U, which means that f is semi-continuous at the identity element eK of K. By [4, Theorem 3.5], f is semi-continuous on K.
Suppose that G, H and K are s-topological groups. Let φ : G →H be a semi-continuous homomorphism, ψ : G →K an irresolute endomorphism such that ker ψ ⊂ ker φ. If ψ is pre-semi-open, then there is a semi-continuous homomorphism f : K → H such that φ = f ∘ ψ .
By Theorem 4.5, there exists a homomorphism f : K → H satisfying φ = f ∘ ψ. We prove that f is semi-continuous. Let V be an open set in H. From φ = f ∘ ψ it follows f←(V) = ψ(φ←(V)). Since, φ is semi-continuous, the set φ←(V) is semi-open in G, and pre-semi-openness of ψ implies that ψ(ψ←(V)) is semi-open, i.e. f←(V) is semi-open in K. This means that f is semi-continuous.
Let (G, *, τG) and (H, ·, τH) be s-topological groups, and f : G → H a homomorphism of G onto H such that for some non-empty open set U →G, the set f(U) is semi-open in H and the restriction f |U : U → f(U) is a pre-semi-open mapping. Then fis pre-semi-open.
We have to prove that if x ∈G and W ∈ SO.(G, x), then f(W)∈SO.(H, f(x). Pick a fixed point y ∈U and consider the mapping
5 Semi-quotients of topologized groups
In this section we apply the construction of sτQ described in Section 3 to topologized groups and establish some properties of their semi-quotients.
If G is a topological group and H a subgroup of G, we can look at the collection G/H of left cosets of H in G (or the collection H\G of right cosets of H in G), and endow G/H (or G\H) with the semi-quotient structure induced by the natural projection p : G → G/H. Recall that G/H is not a group under coset multiplication unless H is a normal subgroup of G.
The following simple lemmas may be quite useful in what follows.
([20]). Let p : G → G/H be a canonicalprojection map. Then for any subset U of G, p←(p(U)) = U * H.
([25]). Let (G, *, τ) be an s-topological group, K an s-compact subset of G, and F a semi-closed subset of G. Then F * K and K * F are semi-closed subsets of G.
([4]). Let (G, *, τ) be an s-topological group. Then each left (right) translation in G is an S-homeomorphism. Moreover they and symmetry mappings are actually semi-homeomorphisim (see [1, Remark 1]).
([26]). If ƒ : X → Y is a semi-continuous mapping and X0 is an open set in X, then the restriction
Let (G, *, τ) be an extremally disconnected irresolute-topological group and H its invariant subgroup. Then ρ :
Let V ⊂ G be semi-open. By the definition of semi-quotient topology, p(V)⊂ G/H is open if and only if p←(p(V))⊂ G is open. By Lemma 5.1 p←(p(V)) = V * H. Since V is semi-open, V * H is semi-open and so p(V)is semi-open. Hence p is pre-semi-open.
Let (G, *, τ) be an extremally disconnected irresolute-topological group, Η its invariant subgroup. Then
First, we observe that sτQ is a topology on G/H. Let x * H, y * H ∈ G/H and let W ⊂G/H be a semi-open neighbourhood of
This just means that
Let (G, *, τ) be an s-topological group and H a subgroup of G. Then for every semi-open set U ⊂G, the set p(U) belongs to sτQ. In particular, if G is extremally disconnected, then p is an s-open mapping from G to (G/H, sτQ).
Let V ⊂G be semi-open. By definition of sτQ, p(V) ∈ sτQ if and only if p←(p(V)) ⊂G is semi-open, i.e. V * H is semi-open in G. But V * H is semi-open in G because V ∈ SO(G) and (G, *, τ)is an s-topological group. Clearly, if sτQ is a topology, the last condition actually says that p is an s-open mapping.
The following theorem is similar to Theorem 5.7.
If Η is an s-compact subgroup of an s-topological group (G, *, τ), then for every semi-closed set F ⊂ G, the set p(G \ F) belongs to sτQ. If sτQ is a topology, then p is an s-perfect mapping.
Let F ⊂G be semi-closed. By Lemma 5.2 the set p←(p(F)) = F * H ⊂G is semi-closed. By definition of sτQ, G/H \ (F * H) ∈ sτQ.
Let now sτQ be a topology on G/H. Take any semi-closed subset F of G. The set F * H is semi-closed in G and F * H = p←(p(F)). This implies, p(F)is closed in the semi-quotient space G/H. Thus p is an s-closed mapping. On the other hand, if z * Η ∈ G/H and p(x) = z * Η for some x ∈ G, then p←(z * H) = p←(p(x)) = x * Η, and by Lemmas 4.3 and 5.3 this set is s-compact in G. Therefore, p is s-perfect.
Let (G, *,τ) be an extremally disconnected s-topological group and Η its s-compact subgroup. If the semi-quotient space (G/H, sτQ) is compact, then G is s-compact.
By Theorem 5.8, the projection p : G → G/H is s-perfect. Then by Theorem 4.4 we obtain that p←(p(G)) = G * H = G is s-compact.
Suppose that (G,*, τ) is an extremally disconnected s -topological group, H an invariant subgroup of G, p:G → (G/H, sτQ) the canonical projection. Let U and V be semi-open neighbourhoods of e in G such that V-1* V ⊂U. Then C1(p(V)) ⊂p(U).
Let p(x) ∈ C1(p(V)). Since V * x is a semi-open neighbourhood of x ∈ G and, by Theorem 5.7, p is s-open, we have that p(V * x)is an open neighbourhood of p(x). Therefore, p(V * x)⋂ p(V)≠ø. It follows that for some a, b ∈ V we have p(a * x) = p(b), that is a * x * h1 = b * h2for some h1, h2∈ Η. Hence,
since a-1 * b ∈ V-1* V ⊂ U and H is a subgroup of G. Therefore, p (x)∈ p(U * H) = U * H * H = U *H = p(U).
Let (G,*, τ) be an extremally disconnected irresolute-topological group and H an invariant subgroup of G. Then the semi-quotient space (G/H, sτQ) is regular.
Let W be an open neighbourhood of p(eG) = Η in G/H. By semi-continuity of p, we can find a semi-open neighbourhood U of eG such that p(U)⊂ W. As G is extremally disconnected and irresolute-topological group, it follows from
If (G, *) is a group, H its subgroup, and a ∈ G, then we define the mapping λa : G/H → G/H by λa(x * Η) = a * (x * Η). This mapping is called a left translation of G/H by a [20].
If (G, *, τ) isanextremally disconnected irresolute-topological group, Η a subgroup of G, and a ∈G, then themapping λa is a semi-homeomorphism and p ∘ ℓa = λa ∘ p holds.
Since G is a group, it is easy to see that λa is a (well defined) bijection on G/H. We prove that λa ∘ p =p ∘ ℓa· Indeed, for each x ∈ G we have (p ∘ ℓa)(x) = p(a * x) = (a*x)* Η = a* (x * Η) = λa(p(x)) = (λa ∘ p)(x). This is required. It remains to prove that λa is irresolute and pre-semi-open.
This follows from the following facts. Let x * Η ∈ G/H. For any semi-open neighbourhood U of eG, p(x *U * H)is a semi-open neighbourhood of x * Η in G/H. Similarly, the set p (a * x * U * H)is a semi-open neighbourhood of a * x * Η in G/H. Since
λa(p(x * U * H)) = p(la(x * U * H)) = p (a * x * U * H),
it follows that λa is a semi-homeomorphism.
A mapping ƒ : X → Y is:
an S-isomorphism if it is an algebraic isomorphism and (topologically) an S -homeomorphism;
a semi-isomorphism if it is an algebraic isomorphism and a semi-homeomorphism.
Let (G,*,τg) and (Η, ·,τη) be extremally disconnected irresolute-topological groups and ƒ : G → H a semi-isomorphism. If G0 is an invariant subgroup of G and H0 = f (G0), then the semi-quotient irresolute-topological groups (G/G0,sτQ) and (H/H0,sτQ) are semi-isomorphic.
Let ρ : G → G/G0, x ↦ x * G0, and π : Η → Η/H0, f(x0)↦ f(x0). H0(x0 ∈ G0) be the canonical projections. Consider the mapping φ : G/Go → H/H0defined by
φ(x * G0) = f(x) · f(G0), x ∈ G, y = f(x).
Then for x1 * G0, x2 * G0 ∈ G/G0 we have
φ(x1 * G0* x2 * G0) = φ(x1 * x2 *G0) = f(x1* x2) · f(G0) = y1 · y2 · H0 = φ(x1 * G0) · φ(x2 * G0),
i.e. φ is a homomorphism. Let us prove that φ is one-to-one. Let x * G0be an arbitrary element of G/G0. Set y = f(x). If φ(x * G0) = H0, then π(y) = H0, which implies x ∈ G0, y ∈ H0, and ker φ = Go. So, φ is one-to-one.
Next, we have φ(x * G0) = y · H0, i.e. φ(p(x)) = π(y) = π(f(x)). This implies φ ∘p = π ∘ f. Since f is a semi-homeomorphism, and p and π are s-open, semi-continuous homomorphisms, we conclude that φ is open and continuous. Hence φ is semi-homeomorphism and a semi-isomorphism.
Let (G, *, τ) be an extremally disconnected irresolute-topological group, H an invariant subgroup of G, M an open subgroup of G, and p G → G/H the canonical projection. Then the semi-quotient group M * H/H is semi-isomorphic to the subgroup p(M) of G/H.
It is clear that M * H = p←(p(M)). As p is s-open and semi-continuous and M is open in τ, the restriction π of p to M * H is an s-open and semi-continuous mapping of M * H onto p(M)by Lemma 5.4. Since M is a subgroup of G and p is a homomorphism it follows that p(M)is a subgroup of G/H, M * H is a subgroup of G, and π: M * H → p(M)is a homomorphism. We have π←(π(eG)) = p←(p(eG)) = H, i.e. ker π = H. It is easy now to conclude that M * H/H and p(M)are semi-isomorphic.
References
[1] Levine N., Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 1963, 70, 36-4110.1080/00029890.1963.11990039Search in Google Scholar
[2] Crossley S.G., Hildebrand S.K., Semi-topological properties, Fund. Math. 1972, 74, 233-25410.4064/fm-74-3-233-254Search in Google Scholar
[3] Bohn E., Lee J., Semi-topological groups, Amer. Math. Monthly, 1965, 72, 996-99810.2307/2313342Search in Google Scholar
[4] Siddique Bosan M., Moiz ud Din Khan, Kocinac Lj. D.R., On s-topological groups, Math. Moravica, 2014, 18:2, 35-4410.5937/MatMor1402035BSearch in Google Scholar
[5] Siab A., Moiz ud Din Khan, Kocinac Lj. D.R., Irresolute-topological groups, math. moravica, 2015, 19:1, 73-8010.5937/MatMor1501073KSearch in Google Scholar
[6] Moiz ud Din Khan and Rafaqat Noreen, On Paratopologized Groups, Analele Universitatii din Oradea - Fasc. Matematica, 2016, Tom XXIII:2, 147-157Search in Google Scholar
[7] Oner T., Kandemir M.B., Tanay B., Semi-topological groups with respect to semi-continuity and irresoluteness, J. Adv. stud. topology, 2013, 4:3, 23-2810.20454/jast.2013.626Search in Google Scholar
[8] Oner T., Ozek A., A note on quasi irresolute topological groups, JLTA, 2016, 5:1, 41-46Search in Google Scholar
[9] Oner T., Ozek A., On semi-topological groups with respect to irresoluteness, Int. J. Recent Sci. Res., 2015, 6:12, 7914-7916Search in Google Scholar
[10] R. Engelking, General Topology, 2nd edition, Heldermann-Verlag, Berlin, 1989Search in Google Scholar
[11] D.R. Anderson, J.A. Jensen, Semi-continuity on topological spaces, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat., 1967, 42,782-783Search in Google Scholar
[12] T.R. Hamlett, A correction to the paper “Semi-open sets and semi-continuity in topological spaces” by Norman Levine, Proc. Amer. Math. Soc, 1975, 49, 458-46010.2307/2040665Search in Google Scholar
[13] Nour T.M., A note on some applications of semi-open sets, Internat. J. Math. Math. Sci., 1998, 21, 205-20710.1155/S0161171298000295Search in Google Scholar
[14] Piotrowski Z., On semi-homeomorphisms, Boll. U.M.I., 1979, 5:16-A, 501-509Search in Google Scholar
[15] Crossley S.G., Hildebrand S.K., Semi-closure, Texas J. Sci., 1971, 22, 99-112Search in Google Scholar
[16] Crossley S.G., Hildebrand S.K., Semi-closed sets and semi-continuity in topological spaces, Texas J. Sci., 1971, 22, 123-126Search in Google Scholar
[17] Corson H.H., Michael E., Metrizability of certain countable unions, Illinois J. Math., 1964, 8, 351-36010.1215/ijm/1256059678Search in Google Scholar
[18] Carnahan D.A., Some properties related to compactness in topological spaces, Ph.D. Thesis, University of Arkansas, 1973Search in Google Scholar
[19] Lee J.P, On semihomeomorphisms, Internat. J. Math. Math. Sci., 1990, 13, 129-13410.1155/S0161171290000163Search in Google Scholar
[20] Arhangel’skii A.V., Tkachenko M., Topological Groups and Related Structures, Atlantis Studies in Mathematics, Vol. 1, Atlantis Press/World Scientific, Amsterdam-Paris, 200810.2991/978-94-91216-35-0_1Search in Google Scholar
[21] Munkres J.R., Topology (Second edition), Prentice Hall, Inc., 2000Search in Google Scholar
[22] Noiri T, On semi-continuous mappings, Atti Accad. Naz. Lincei, Ser., 1973, VIII 54:2, 210-214Search in Google Scholar
[23] Pipitone V, Russo G., Spazi semiconnessi e spazi semiaperti, Rend. Circ. Mat. Palermo, 1975, 24:3, 273-28510.1007/BF02843735Search in Google Scholar
[24] Njåstad O., On some classes of nearly open sets, Pacific J. Math., 1965, 15,961-97010.2140/pjm.1965.15.961Search in Google Scholar
[25] Siddique Bosan M., Moiz ud Din Khan, A note on s-topological groups, Life Science Journal, 2014, 11:7s, 370-374Search in Google Scholar
[26] Noiri T, On semi-continuous mappings, Atti.Lin.Rend.Sc.Fis.Mat.Natur., 1973, 8:54, 210-215.Search in Google Scholar
© Khan et al.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.