Semi-quotient mappings and spaces

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.

the pre-image of B. By Cl.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 e G denotes its identity element, and for a given x 2 G,`x W G ! G, y 7 ! x ı y, and r x W G ! G, y 7 ! y ı x, denote the left and the right translation by x, respectively. The operation we call the multiplication mapping m W G G ! G, and the inverse operation x 7 ! x 1 is denoted by i .
Recall that a set U X is a semi-neighbourhood of a point  [13].
A mapping f W X ! Y between topological spaces X and Y is called: s-perfect if it is semi-continuous, s-closed, surjective, and f .y/ is s-compact relative to X , for each y in Y .
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 2 G to the coset in G=H containing g) will be denoted by p. A mapping f W G ! H between two topological groups is called a topological isomorphism if f is an algebraic isomorphism and a topological homeomorphism.

Semi-quotient mappings
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. Example 3.2. Let X D Y D f1; 2; 3g and let X D f;; X; f1g; f2g; f1; 2g; f1; 3gg and Y D f;; Y; f1g; f2g; f1; 2gg be topologies on X and Y . Let f W X ! Y be defined by f .x/ D x, x 2 X . Since Y X , the mapping f is continuous, hence semi-continuous. On the other hand, this mapping is not semi-quotient because f .f1; 3g/ is semi-open in X although f1; 3g is not open in Y .
The mapping f is not a quotient 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.
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 D f2; 4g.
To see when the restriction of a semi-quotient mapping is also semi-quotient we will need the following simple but useful lemmas.  . Let X be a topological space, X 0 a subspace of X . If A 2 SO.X 0 /, then A D B \ X 0 , for some B 2 SO.X /.

Now we have this result.
Theorem 3.8. Let f W X ! Y be a semi-quotient mapping and let A be a subspace of X saturated with respect to f , and let g W A ! f .A/ be the restriction of f to A. Then: This means that V is open in Y and thus in f .A/. This completes the proof that g is a semi-quotient mapping.
Other part is the same as in (a).
As a complement to Proposition 3.4 we have the following two theorems.
Then the mapping g ı f W X ! Z is semi-quotient if and only if g is a quotient mapping.
Proof. If g is a quotient mapping, then g ı f is semi-quotient as the composition of a semi-quotient and a quotient mapping (Proposition 3.4).
Conversely, let g ı f be semi-quotient. We have to prove that 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 W X ! Y be a mapping. Define It is easy to see that the family s Q ia a generalized topology on Y (i.e. ; 2 s Q and union of any collection of sets in s Q is again in s Q ) generated by f ; we call it the semi-quotient generalized topology. But s Q 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 s Q is the finest topology on Y such that f W X ! .Y; / is semi-continuous. In fact, f W X ! .Y; s Q / is a quotient mapping in this case. In particular, let be an equivalence relation on X . Let p W X ! X= be the natural (or canonical) projection from X onto the quotient set X= : for each x in X , p sends x to the equivalence class .x/. Then the family s Q generated by p is a generalized topology on the quotient set Y = , and a topology when X is extremally disconnected. This topology will be called the semi-quotient topology on X= . Observe, that we forced the mapping p to be semicontinuous, that is semi-quotient.
This kind of construction will be applied here to topologized groups: to s-topological groups and irresolutetopological 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.

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.

Definition 4.1 ([3]
). An s-topological group is a group .G; / with a topology such that for each x; y 2 G and each neighbourhood W of x y 1 there are semi-open neighbourhoods U of x and V of y such that U V 1 W .   ; The following results are related to s-topological groups, and they are generalizations of some results for topological groups.

Semi-quotients of topologized groups
In this section we apply the construction of s Q 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 nG of right cosets of H in G), and endow G=H (or GnH ) with the semi-quotient structure induced by the natural projection p W 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.   [25]). Let .G; ; / 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.

Lemma 5.3 ([4]
). Let .G; ; / be an s-topological group. Then each left .right/ translation in G is an Shomeomorphism. Moreover they and symmetry mappings are actually semi-homeomorphisim (see [1,Remark 1]). Lemma 5.4 ([26]). If f W X ! Y is a semi-continuous mapping and X 0 is an open set in X , then the restriction f j X 0 W X 0 ! Y is semi-continuous. The following theorem is similar to Theorem 5.7.
Theorem 5.8. If H is an s-compact subgroup of an s-topological group .G; ; /, then for every semi-closed set F G, the set p.G n F / belongs to s Q . If s Q is a topology, then p is an s-perfect mapping.
Proof. Let F G be semi-closed. By Lemma 5.2 the set p .p.F // D F H G is semi-closed. By definition of s Q , G=H n .F H / 2 s Q .
Let now s Q 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 D 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 H 2 G=H and p.x/ D z H for some x 2 G, then p .z H / D p .p.x// D x H , and by Lemmas 4.3 and 5.3 this set is s-compact in G. Therefore, p is s-perfect.
Corollary 5.9. Let .G; ; / be an extremally disconnected s-topological group and H its s-compact subgroup. If the semi-quotient space .G=H; s Q / is compact, then G is s-compact.
Proof. By Theorem 5.8, the projection p W G ! G=H is s-perfect. Then by Theorem 4:4 we obtain that p .p.G// D G H D G is s-compact.  If .G; / is a group, H its subgroup, and a 2 G, then we define the mapping a W G=H ! G=H by a .x H / D a .x H /. This mapping is called a left translation of G=H by a [20].
Theorem 5.12. If .G; ; / is an extremally disconnected irresolute-topological group, H a subgroup of G, and a 2 G, then the mapping a is a semi-homeomorphism and p ı`a D a ı p holds.
Proof. Since G is a group, it is easy to see that a is a (well defined) bijection on G=H . We prove that a ıp D pı`a. it follows that a is a semi-homeomorphism.
Definition 5. 13. A mapping f W 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.
Theorem 5.14. Let .G; ; G / and .H; ; H / be extremally disconnected irresolute-topological groups and f W G ! H a semi-isomorphism. If G 0 is an invariant subgroup of G and H 0 D f .G 0 /, then the semi-quotient irresolute-topological groups .G=G 0 ; s Q / and .H=H 0 ; s Q / are semi-isomorphic.
Next, we have '.x G 0 / D y H 0 , i.e. '.p.x// D .y/ D .f .x//. This implies ' ı p D ı f . Since f is a semi-homeomorphism, and p and are s-open, semi-continuous homomorphisms, we conclude that ' is open and continuous. Hence ' is semi-homeomorphism and a semi-isomorphism.