Extension closed properties on generalized topological groups

In this paper, we continue to investigate some important results in generalized topological groups and we prove extension closed property for connectedness, compactness, and separability of generalized topological groups. Last, we define generalized topological group actions on generalized topological spaces and we establish a homeomorphism between action group and action space.


Introduction
In [2], Császár introduced and extensively studied the notion of generalized open sets discarding finite intersection axiom from the general topology. Since then he and many other authors in the literature have shown that important properties and results still hold, with some or no modification.
In [8], we defined the generalized topological group structure and we proved some basic results. Especially, we examined generalized connectedness property in [8].
In [9], we defined the ultra Hausdorff property of spaces and we gave some basic characterizations and we investigated the relation between generalized compactness and ultra Hausdorfness.
In this paper, we continue to investigate some important results in generalized topological groups and prove extension closed property for connectedness, compactness, first countability and separability of generalized topological groups. In the last section, we define generalized topological group actions on generalized topological spaces and we establish a homeomorphism between action group and action space.
Proof Let U be a G-open neighborhood of e. For each n ∈ N, we denote by U n the set of elements of the form u 1 . . . u n , where each u i ∈ U . Let W := n∈N U n . Since each U n is G-open, we have that W is a G-open set. We now see that it is also G-closed. Let g be an element of generalized closure of W . That is, g ∈ Cl G W . Since gU −1 is a G-open neighborhood of g, it must intersect W . Thus, let h ∈ W ∩ gU −1 .
Since h ∈ gU −1 , then h = gu −1 for some elements u ∈ U . Since h ∈ W , then h ∈ U n for some n ∈ N, i.e., h = u 1 . . . u n with each u i ∈ U . We then have g = u 1 . . . u n u, i.e., g ∈ U n+1 ⊆ W . Hence, W is G-closed. Since G is G-connected and W is G-open and G-closed, we must have W = G. This means that G is generated by U . Proof Assume that the subgroup K is non-trivial. Take an arbitrary element x ∈ K distinct from the identity e of G. Since the group is discrete, we can find a G-open neighborhood U of x in G such that U ∩ K = {x}. It follows from the G-continuity of the multiplication in G and the obvious equality exe = x that there exists Since the group G is G-connected, Lemma 2.8 implies that the sets V n , with n ∈ N, cover the group G. Therefore, every element g ∈ G can be written in the form g = y 1 ...y n , where y 1 , ..., y n ∈ V and n ∈ N. Since x commutes with every element of V , we have gx = y 1 · · · y n x = y 1 · · · x y n = · · · = y 1 x · · · y n = x y 1 · · · y n = xg.
We have proved that the element x ∈ K is in the center of the group G. Since x is an arbitrary element of K , we conclude that the center of G contains K .
By Theorem 2.9, we have the following result.

Theorem 2.11
If H is a G-dense subgroup of a G-connected G-topological group, then every G-neighborhood U of the identity element in H algebraically generates the group H .

Definition 2.12
A space X is called G-resolvable if there exists G-dense disjoint subsets A and B of X .
Let G be a G-topological group. Since G is homogeneous and the union of resolvable spaces is again resolvable then we have the following results.
Proof (i) We reach the aim by generalized subspace topology. (ii) It is coming from the first result (i) since closure of a generalized topological subgroup is subgroup. (iii) It is coming from (i) and (iii). Theorem 2.14 Let f : G → H be a G-continuous mapping of G-topological spaces. If G is G-compact and H is G-ultra Hausdorff and G-normal, then f is G-closed.
By assumption, H is ultra Hausdorff and normal, so f (K ) is G-closed.

Theorem 2.16 Let f : G → H be a G-continuous onto homomorphism of G-topological groups. If G is G-compact and H is G-ultra Hausdorff and normal, then f is G-open.
Proof By Theorem 2.14, the mapping f is G-closed, and hence it is quotient.
. Equality is evident.

Theorem 2.19 Let H be a G-closed invariant subgroup of a G-topological group G. If H and G/H are G-connected, then so is G.
Proof Suppose that H and G/H are G-connected and f : G → {0, 1} be an arbitrary G-continuous map. We have to show that f is constant. The restriction of f to H must be constant and since each coset g H is G-connected, f must be constant on g H as well taking value f (g). Thus, we have a well-defined map f : G/H → {0, 1} such thatf • π = f . By the fundamental property of quotient spaces, it follows thatf is G-continuous and so must be constant since G/H is G-connected. Hence, f is also constant and we conclude that G is G-connected.

G-topological group actions on G-topological spaces
In this section, we will introduce G-topological Group Actions on G-topological spaces and we want to improve some results from topological group action theory.
For a point x ∈ X , the set G(x) = {gx : g ∈ G} is called the orbit of x. If this map is G-continuous, then the action is said to be G-continuous. The space X , with a given Gcontinuous action of G on X , is called G-space.
Proof It is sufficient to verify that the images under θ of the elements of some base for G × X are G-open in X.

Proposition 3.3
The G-continuity of an action θ : G × X → X of a G-topological group G with identity e on a space X is equivalent to the G-continuity of θ at the points of the set {e} × X ⊂ G × X.
Proof Let g ∈ G and x ∈ X be arbitrary and U be a neighborhood of gx in X . Since θ h is a homeomorphism of X for each h ∈ G, the set V = θ g−1 (U ) is a neighborhood of x in X . By the G-continuity of θ at (e, x), we can find a neighborhood O of e in G and a neighborhood W of x in X such that hy ∈ V for all h ∈ O and y ∈ W . Clearly, if h ∈ O and y ∈ W , then (gh)(y) = g(hy) ∈ gV = θ g (V ) = U . Thus, ky ∈ U , for all k ∈ gO and all y ∈ W , where O = gO is a neighborhood of g in G. Hence, the action θ is G-continuous.
Here are some examples of G-continuous actions of G-topological groups.
Example 3.4 Any G-topological group G acts on itself by left translations, i.e., θ(x, y) = x y for all x, y ∈ G. The G-continuity of this action follows from the G-continuity of the multiplication in G. Example 3.5 Let G be a topological group, H a G-closed subgroup of G, and let G/H be the corresponding left coset space. The action φ of G on G/H , defined by the rule φ(g, x H) = gx H , is G-continuous. Indeed, take any y 0 ∈ G/H , and fix a G-open neighborhood O of y 0 in G/H . Choose x 0 ∈ G such that π(x 0 ) = y 0 , where π : G → G/H is the G-quotient mapping. There exists G-open neighborhoods U and V of the identity e in G such that π(U x 0 ) ⊂ O and V 2 ⊂ U . Clearly, W = π (V x 0 ) is G-open in G/H and y 0 ∈ W. By the choice of U and V , if g ∈ V and y ∈ W , then φ(g, y) ∈ O. Indeed, take x 1 ∈ V x 0 with π(x 1 ) = y. Then, y = x 1 H and φ(g, y) = gx 1 Suppose that a G-topological group G acts continuously on a space X and that Y = X/G is the corresponding orbit set. Let Y carry the quotient G-topology generated by the orbital projection π : G-open in X ). The G-topological space X/G so obtained is called the orbit space or the orbit space of the G-space X . The orbital projection is always a G-open mapping:

Proposition 3.6 If θ : G × X → X is a G-continuous action of a topological group G on a space X , then the orbital projection π : X → X/G is G-open.
Proof For a G-open set U ⊂ X , consider the set π −1 π(U ) = GU . Every left translation θ g is a homeomorphism of X onto itself, so the set

Theorem 3.7 If a G-compact G-topological group H acts continuously on an ultra-G-Hausdorff space X , then the orbital projection π : X → X/H is a G-open and G-perfect mapping.
Proof Let Y = X/H . If y ∈ Y, choose x ∈ X such that π(x) = y and note that π −1 (y) = H x is the orbit of x in X . Since the mapping of H onto H x assigning to every g ∈ H , the point gx ∈ X is G-continuous, the image H x of the G-compact group H is alsoG-compact. Hence, all fibers of π are G-compact.
To verify that the mapping π is G-closed, let y ∈ Y and x ∈ X be as above, and let O be a G-open set in X containing π −1 (y) = H x. Since the action of H on X is G-continuous, we can find, for every g ∈ H , G-open neighborhoods U g g and V g x in H and X , respectively, such that U g V g ⊂ O. By the G-compactness of H and of the orbit H x, there exists a finite set F ⊂ H such that H = g∈F U g and H x ⊂ g∈F gV g . Then, V = g∈F V g is a G-open neighborhood of x in X , and we claim that H V ⊂ O. Indeed, if h ∈ H and z ∈ V , then h ∈ U g , for some g ∈ F, so that hz ∈ U g V ⊂ U g V g ⊂ O. Thus, W = π(V ) is a G-open neighborhood of y in Y , and we have π −1 π(V ) = H V ⊂ O. Hence, the mapping π is G-closed. Finally, π is G-open, by Proposition 3.6. g, x)), i.e., g f (x) = f (gx), for all g ∈ G and all x ∈ X . Clearly, f is G-equivariant if and only if the diagram below commutes, where F = id G × f is the product of the identity mapping id G of G and the mapping f .

Example 3.9
Let H be a G-closed subgroup of a G-topological group G, and Y = G/H be the left coset space. Denote by θ G the action of G on itself by left translations, and by θ Y the natural G-continuous action of G on Y . Then, the quotient mapping π : G → G/H defined by π(x) = x H for each x ∈ G is equivariant. Indeed, the equality g(π(x)) = gx H = π(gx) holds for all g, x ∈ G. Equivalently, the diagram is commutative, Let η = {X i : i ∈ I } be a family of G-spaces. Then, the product space X = i∈I X i , if X is ultra-G-Hausdorff, is a G-space. To define an action of G on X , take any g ∈ G and any x = (x i ) i∈I ∈ X , and put gx = (gx i ) i∈I . Thus, G acts on X coordinatewise. The following result guarantees the G-continuity of this action.

Proposition 3.10
The coordinatewise action of G on the product X = i∈I X i of G-spaces is G-continuous, i.e., X is a G-space, if X is ultra-G-Hausdorff.
It follows immediately from the definition of the sets U and W that U W ⊂ O. Therefore, the action of G on X is G-continuous.

Theorem 3.11 Let G act G-continuously on X and suppose that both G and X/G are G-connected, then X is G-connected.
Proof Suppose X is the union of two disjoint nonempty G-open subsets U and V . Now π(U ) and π(V ) are G-open in X/G. Since X/G is G-connected, π(U ) and π(V ) cannot be disjoint. If π(x) ∈ π(U ) ∪ π(V ), then Then, we can give the relationship between group actions and separation axiom in the following.

Theorem 3.12 If X is G-compact topological group and G a G-closed subgroup acting on X by left translation, then X/G is G-regular, and so X/G is G-Hausdorff.
Proof Since G is a G-closed subgroup and the left translation map L x : X → X is a G-homeomorphism then π −1 π(x) = xG = L x (G) is G-closed. Thus, every point π(x) of X/G is G-closed, and it follows that X/G is G-T 1 space. Now we will show that for a G-closed subset F of X/G and a point p / ∈ F, there are G-open sets U, V satisfying p ∈ U, F ⊂ V, U ∩ V = ∅. Since X acts transitively on X/G, we may suppose that p is element of the class eG = G of the identity element e. Since F is G-closed, there exists a G-open set U 0 such that F ∩ U 0 = ∅ and p ∈ U 0 . From the continuity of group action of X , there is a G-open set W such that e ∈ W and Since π is G-open map, both the sets U = π(W ) and V = π(W π −1 (F)) are G-open and such that p ∈ U and F ⊂ V . Last, we will show that U ∩ V = ∅ by contradiction. So assume that there exists y ∈ U ∩ V . Then, there exists x 1 , x 2 ∈ W and x ∈ π −1 (F) such that y = π(x 2 ) = π(x 1 x). Thus, we have g ∈ G such that To give the last important result, we need the following definitions.

Definition 3.13
Let G act on the generalized topological space X . Then, for a point x of X the set is a subgroup of G and it is called stabilizer of x in G.
We should note that G x is G-closed since the singleton set {x} is G-closed. Definition 3.14 Let G act on the generalized topological space X . Then, for a point x of X , we define a map By continuity of action μ x is G-continuous. Obviously, we have the following facts.
(i) μ x is surjective iff G acts transitively on X .
(ii) x∈X G x = {e} iff G acts effectively on X .
Now we are ready to prove the following result.

Theorem 3.15
If G is G-compact, X ultra G-Hausdorff, and if G acts transitively on X , then X is homeomorphic to the orbit space G/G x for any x ∈ X.
Proof Let X be a homogeneous G-space of G. First, we claim that μ x induces a bijection h x : G/G x → X such that μ x = h x • π x , where π x : G → G/G x is orbital projection. We have π(g 1 ) = π(g 2 ) iff g 1 −1 g 2 ∈ G x iff g 1 −1 g 2 x = x iff μ(g 1 ) = μ(g 2 ). The equality h x (π x (g)) = μ x (g) determines the injection h x . On the other hand, from the first fact above μ x is surjective implies h x is surjective.
Next, we claim that for g ∈ G, x ∈ X and y = gx, the diagram is commutative, where G y = gG x g −1 and A g is a homeomorphism given by A g (g G x ) = (gg g −1 )G y . Indeed, for g 1 ∈ G, L g (h x (π x (g 1 ))) = L g (μ x (g 1 )) = gg 1 x = gg 1 g −1 y = h y (A g (π y (g 1 ))). It implies that the diagram is commutative. Clearly, we have G y = A g G x = gG x g −1 .