Semitopological Homomorphisms
ANNAGIORDANOBRUNO(*)
ABSTRACT- Inspired by an analogous result of Arnautov about isomorphisms, we prove that all continuous surjective homomorphisms of topological groups f :G!H can be obtained as restrictions of open continuous surjective homo- morphisms fe:Ge!H, where Gis a topological subgroup of G. In case thee topologies onGandH are Hausdorff andH is complete, we characterize the continuous surjective homomorphisms f :G!H when Ghas to be a dense normal subgroup ofG.e
We consider the general case whenGis requested to be a normal subgroup ofG,e that is whenf issemitopologicalÐ Arnautov gave a characterization of semi- topological isomorphisms internal to the groupsGandH. In the case of homo- morphisms we define new properties and consider particular cases in order to give similar internal conditions which are sufficient or necessary for f to be semitopological. Finally we establish a lot of stability properties of the class of all semitopological homomorphisms.
1. Introduction
In [2, Theorem 1] Arnautov showed that for every continuous iso- morphismf :(G;t)!(H;s) of topological groups, there exist a topological group (G;ee t) containingGas a topological subgroup and an open continuous homomorphism ef :(G;e et)!(H;s) extending f. Moreover he noted that such a pair (G;e ef) need not always exist under the additional assumption thatGis a normal subgroup ofG. So in [2, Definition 2] the author defined ae continuous isomorphism f :(G;t)!(H;s) of topological groups to be semitopological if there exist a topological group (G;e et) containingGas a topological normal subgroup and an open continuous homomorphism
(*) Indirizzo dell'A.: Dipartimento di Matematica e Informatica, UniversitaÁ di Udine, Via delle Scienze, 206 - 33100 Udine, Italy.
E-mail: anna.giordanobruno@dimi.uniud.it
2000 Mathematics Subject Classification: Primary 22A05, 54H11; Secondary:
18A20, 20F38, 20K45.
ef :(G;e et)!(H;s) extendingf (the counterpart of this definition for topo- logical rings was given in [1]).
Iff :G!His a group homomorphism, denote byGf thegraphoff, that is the subgroupGf f(g;f(g)):g2GgofGH. Using explicitly the graph of a continuous surjective group homomorphismf :G!Hwe can prove [2, Theorem 1] in a much simpler way and even generalize it weakening the hypothesis onf(in our theoremfis a homomorphism while in [2, Theorem 1]f was an isomorphism) and achievingGto be aclosed subgroup ofG, in case the topology on the codomain is Hausdorff.e
THEOREM1.1. Let G, H be topological groups and f :(G;t)!(H;s)a continuous surjective homomorphism. Then there exist a topological group(G;e et)containing(G;t)as a topological subgroup and an open con- tinuous homomorphismef :(G;ee t)!(H;s)such thatefj33Gf . If (H;s)is Hausdorff, then(G;t)is closed in(G;ee t).
PROOF. DefineGe GHandetts. It follows from [4, Remark 2.12] that (G;t) is topologically isomorphic to (Gf;etj33Gf) (the topological isomorphism isj:G!Gf defined byj(x)(x;f(x)) for everyx2G). The canonical projection p2:(G;ee t)!(H;s) is an open continuous homo- morphism extending f and so we take ef p2. If (H;s) is Hausdorff, (Gf;etj33Gf) is a closed subgroup of (G;ee t) by the closed graph theorem. p Inspired by this result we give the counterpart of [2, Definition 2] for continuous surjective homomorphisms:
DEFINITION1.2. A continuous surjective homomorphism f :(G;t)!
!(H;s) of topological groups issemitopologicalif there exist a topological group (G;ee t) containingG as a topological normal subgroup and an open continuous homomorphismef :(G;e et)!(H;s) extendingf.
It is obvious that all open continuous surjective group homomorphisms are semitopological.
Theorem 1.1 shows that every continuous surjective homomorphism of topological groups f :G!H is the restriction of an open continuous surjective homomorphism of topological groupsef :Ge !H, where Gi s a topological subgroup ofG. As noted before not every continuous surjectivee group homomorphism is the restriction of an open continuous surjective group homomorphism to a normal topological subgroup of the domain, i.e.
not all continuous surjective group homomorphisms are semitopological.
Consequently the characterization of semitopological homomorphisms can also be viewed as the study of the restrictions of open continuous surjective homomorphisms of topological groupsef :Ge!Hto normal subgroupsGof Ge such that ef(G)H. From this point of view semitopological homo- morphisms are strictly related to the open mapping theorem and its generalizations, which are studied by a lot of authors.
Let S be the class of all semitopological homomorphisms and Si the class of all semitopological isomorphisms. ObviouslySi S.
To formulate the main theorem of [2] characterizing semitopological isomorphisms, we recall the following concept: for a neighborhoodUof the neutral elementeGof a topological groupGcall a subsetMofG (left) U- thin if T
fx 1Ux:x2Mg is still a neighborhood of eG. We give some properties ofU-thin sets in §3.
For a topological group (G;t) we denote by V(G;t)(eG) the filter of all neighborhoods ofeGi n (G;t).
THEOREM 1.3 [2, Theorem 4]. Let (G;t);(H;s)be topological groups and f :(G;t)!(H;s) a continuous isomorphism. Then f is semi- topological if and only if for every U2 V(G;t)(eG)
(a) there exists V 2 V(H;s)(eH)such that f 1(V)is U-thin;
(b) for every g2G there exists Vg2 V(H;s)(eH) such that [g;f 1(Vg)]U.
In this case of continuous isomorphisms it is possible to consider without loss of generality the same group G as domain and codomain, endowed with two different group topologies ts and as continuous isomorphism the identity map 1G of G. In fact, if f :(G;t)!(H;h) i s a continuous isomorphism of topological groups, then the topology sf 1(h) onGis coarser thantand so 1G:(G;t)!(G;s) is a continuous isomorphism and (G;s) is topologically isomorphic to (H;h). In particular 1G:(G;t)!(G;s) is semitopological if and only if f :(G;t)! (H;h) i s semitopological. So the following is an equivalent form of Theorem 1.3.
THEOREM1.4. Let G be a group andtsgroup topologies on G. Then 1G:(G;t)!(G;s)is semitopological if and only if for every U2 V(G;t)(eG)
(a) there exists V 2 V(G;s)(eG)such that V is U-thin;
(b) for every g2G there exists Vg2 V(G;s)(eG)such that[g;Vg]U.
The aim of this paper is to generalize these and most of the remaining results of [2] to semitopological homomorphisms. To find a complete
characterization of semitopological homomorphisms has turned out to be a non-trivial problem.
However we find a complete characterization in a particular case, which is interesting on its own: as said before semitopological homomorphisms can also be viewed as restrictionsf :G!Hof open continuous surjective homomorphisms of topological groups ef :Ge !H where G is a normal subgroup ofGe andf(G)H. In this setting one can askGto be adense normal subgroup ofG:e
DEFINITION1.5. Let (G;t);(H;s) be topological groups. A continuous surjective homomorphismf :(G;t)!(H;s) i sd-semitopologicalif there exist a topological group (G;e et) containingGas a dense normal topological subgroup and an open continuous homomorphism ef :(G;ee t)!(H;s) ex- tendingf.
In §2 we study d-semitopological homomorphisms. In this case con- sidering Hausdorff group topologies simplifies the problem and in fact, with the additional assumption that the codomain is complete, we find a complete characterization of d-semitopological homomorphisms in Theo- rem 1.6. For a subgroupHof a groupG,NG(H) fx2G:xHHxgis the normalizer ofHinG. IfGis a Hausdorff topological group, we denote byGthe two-sided completion ofG.
THEOREM 1.6. Let f :G!H be a continuous surjective homo- morphism, where G is a Hausdorff group and H is a complete group, and let f :G!HH be the extension of f to the completions. Then the fol- lowing conditions are equivalent:
(a) f is d-semitopological;
(b) fj33N
G(G):NG(G)!H is open;
(c) NG(G)\kerf is dense inkerf .
In §4 we find necessary and sufficient conditions for a continuous surjective group homomorphism to be semitopological. To do this we in- troduce new properties. For example we define A-open and strongly A- open surjective group homomorphisms (see Definitions 4.10 and 4.16 re- spectively) and without any recourse to Arnautov's main theorem (i.e.
Theorem 1.3) we prove that
strongly A-open)semitopological)A-open
for continuous surjective group homomorphisms (see Theorems 4.12
and 4.20). Since strongly A-open coincides with A-open for continuous group isomorphisms as well as with conditions (a) and (b) of Theorem 1.3, Theorem 1.3 is an immediate corollary of this result. In §4 we de- fine also A-open and strongly A-open surjective group homomorph- isms (see Definitions 4.16 and 4.24 respectively). Also these properties are equivalent to semitopological for continuous group isomorphisms.
The main relations among new and already defined properties for continuous surjective group homomorphisms are the following:
1
Moreover in Theorem 4.29 we prove that all these four properties are equivalent to semitopological for a continuous surjective homomorph- ism of topological groups f :G!Hsuch that kerf i s contai ned i n the closure of eG in G. This assumption is quite strong, but continuous isomorphisms satisfy it and so we find again Theorem 1.3 as a trivial corollary.
In order to find some property equivalent to semitopological, we ana- lyze the relations among all the properties in (1). Some questions about these relations remain open and a positive answer to them would bring a more precise description of semitopological homomorphisms and their properties.
As a particular case of Theorem 1.3, a continuous isomorphism of topological groups f :(G;dG)!(H;s), wheredG is the discrete topology on G, is semitopological if and only if the centralizer cH(h)
fx2H:[x;h]eHgofhinHiss-open for everyh2H[2, Corollary 5]
(see Corollary 5.2). In §5 we consider this particular case for homo- morphisms, but we also prove some results in case the topology on the codomain is indiscrete.
Moreover we consider continuous surjective group homomorphisms with the properties in (1) in case the topologies on the domain and on the codomain are very close to be discrete or indiscrete.
In some of these particular cases we find examples and counter- examples clarifying the relations among the properties in (1). This can give an indication of which direction we have to take.
Thanks to the characterization of Theorem 1.3 Arnautov proved thatSi
is stable under taking subgroups, quotients and products:
THEOREM 1.7. Let (G;t);(H;s) be topological groups, f :(G;t)!
!(H;s)a semitopological isomorphism and A a subgroup of G. Then:
(a) [2, Theorem 7]fj33A:(A;tj33A)!(f(A);sj33f(A))is semitopological;
(b) [2, Theorem 8]in case A is normal and q:G!G=A, q0:H!
!H=f(A) are the canonical projections, f0:(G=A;tq)!(H=f(A);sq), defined by f0(q(g))q0(f(g))for every g2G, is semitopological.
THEOREM1.8 [2, Theorem 9]. Letf(Gi;ti):i2Ig,f(Hi;si):i2Igbe families of topological groups and fi:(G;ti)!(H;si) a semitopological isomorphism for every i2I. Then Q
i2Ifi:Q
i2I(G;ti)!Q
i2I(H;si) is semitopological.
In §6 we prove (categorical) stability properties of the larger classS and of the new properties we introduce, extending substantially Theorems 1.7 and 1.8. In particular in Theorem 6.15(a) we prove that alsoSis stable under taking products:
If ffi:i2Ig is a family of continuous surjective group homo- morphisms, thenQ
i2Ifi2 Sif and only iffi2 Sfor alli2I.
For (strongly) A-open and (strongly) A-open we prove that also the converse implication holds. Consequently, since they coincide with semi- topological for continuous group isomorphisms, we obtain as a bonus a new result, namely that the converse implication in Theorem 1.8 holds true.
About categorical properties, we prove thatSiis closed under pullback along continuous injective homomorphisms and that the class of all A-open continuous surjective homomorphisms is stable under pushout with re- spect to open continuous surjective group homomorphisms (see Theorems 6.3 and 6.14 respectively). In these terms Theorem 1.7 says thatSiis closed under pullback with respect to topological embeddings and under pushout with respect to open continuous surjective homomorphisms and so it be- comes a corollary of our results.
As noted in [2]Siis not closed under taking compositions (see Example 5.18). Hence also (strongly) A-open and (strongly) A-open are not stable under taking compositions because they are equivalent to semitopological for continuous group isomorphisms. However we prove some results about composition of semitopological homomorphisms and about composition of group homomorphisms which are (strongly) A-open and (strongly) A-open.
In Corollary 6.5 we find a characterization for an A-open continuous surjective group homomorphism in terms of its algebraically associated isomorphism:
Iff:(G;t)!(H;s)is a continuous surjective homomorphism and q:G!G=kerf the canonical projection, then by the first homo- morphism theorem for topological groupsf0:(G=kerf;tq)!(H;s), defined byf0(q(g))f(g)for everyg2G, is a continuous isomor- phism. Thenf is A-open if and only iff0is semitopological.
Note that the first homomorphism theorem for topological groups says that in this casef is open if and only if f0 is open. In analogy our result states that f is A-open if and only iff0 is A-open (for continuous group isomorphisms A-open coincides with semitopological).
One of the most interesting results is Proposition 6.10 which shows that the class of strongly A-open continuous surjective group homomorphisms is left cancelable. As a consequence we get that the class Si is left can- celable (see Theorem 6.11).
In §7 we sum up in a more detailed diagram the relations among the properties in (1) for continuous surjective group homomorphisms. We also collect there all open questions. Moreover we remind the open problems left in [2] about semitopological isomorphisms, that we discuss in a dif- ferent paper [3].
Notation and terminology.
LetGbe a group andx;y2G. We denote by [x;y] the commutator ofx and y in G, that is [x;y]xyx 1y 1. If H and K are subsets of G, let [H;K] h[h;k]:h2H;k2Ki. In caseH fxgfor somex2G, we write [x;K]. MoreoverG0[G;G] is the subgroup ofGgenerated by all com- mutators of elements ofG. We denote the center ofGbyZ(G).
The functionD:G!GGis defined byD(g)(g;g) for everyg2G.
If H is another group, we call p1:GH!G and p2:GH!H the canonical projections on the first and the second component respectively.
Lettbe a group topology onG. IfXis a subset ofG,Xt stands for the closure ofXi n (G;t). IfNis a normal subgroup ofGandq:G!G=Nis the canonical projection, then tq is the quotient topology of t on G=N.
MoreoverNtdenotes the subgroupfeGgt. The discrete topology onGisdG and the indiscrete topology onGisiG. For a Hausdorff topological groupG we denote byGthe completion ofG. A groupGis complete ifGG. Let us note that in particular every complete group is Hausdorff.
For undefined terms see [6, 7].
2. Dense extensions.
We discuss here the problem of characterizing d-semitopological group homomorphisms in the class of Hausdorff topological groups.
DEFINITION 2.1. Let (G;t);(H;s) be topological groups and f :(G;t)!(H;s) a continuous surjective homomorphism. Call the topolo- gical group (G;e et) ad-extensionoff :(G;t)!(H;s) i f (G;e et) contains (G;t) as a dense normal topological subgroup and there exists an open continuous homomorphism ef :(G;ee t)!(H;s) extending f. The homomorphism ef is calledassociatedto the d-extension (G;e et).
Let (G;t), (H;s) be Hausdorff groups. Every continuous surjective homomorphismf :(G;t)!(H;s) can be extended to a continuous homo- morphism of the completionsf :(G;t)!(H;s). Iff is open, thenfis open.
The following fact is due to Grant.
FACT2.2 [5, Lemma 4.3.2]. Let G;H be Hausdorff topological groups, f :G!H an open continuous surjective homomorphism and N a dense subgroup of G. Then fj33N:N!f(N)is open if and only if N\kerf is dense inkerf .
CLAIM 2.3. Let G;H be Hausdorff topological groups, f :G!H a continuous surjective homomorphism and let f :G!H be the extension of f to the completions. If f is d-semitopological, then NG(G)\kerf is dense inkerf .
PROOF. LetGe be a topological group such thatGis a dense normal subgroup of Ge and ef :Ge !H an open continuous surjective homo- morphism. Being Gdense in G, by the universal property of the com-e pletion we can identifyGewith a subgroup ofG; i n parti cularGandGehave the same completion G. Moreover, sinceGi s normal i n G,e Ge NG(G).
For the property of the completion, there exists a continuous homo- morphismf :G!Hextendingef (and so extendingf). Beingef open and Gedense inG,Hdense inH, sofis open as well. By Fact 2.2NG(G)\kerf
i s dense i n kerf. p
Applying this claim we can now prove Theorem 1.6:
PROOF OFTHEOREM1.6. (a))(c) is Claim 2.3 and (b))(a) is obvious becauseGis dense inGandG/NG(G).
(b),(c) follows from Fact 2.2 sinceHH. p
COROLLARY2.4. If there exists a d-extension of a continuous surjec- tive homomorphism of topological groups f :G!H, whereGis Haus- dorff and H complete, then there exists a maximal one, namely NG(G) with the topology inherited fromG.
Theorem 1.6 can be written in the following equivalent form:
Let K;H be complete groups and h:K!H a continuous open sur- jective homomorphism. If G is a dense subgroup of K such that h(G)H, then the following conditions are equivalent:
(a) hj33G:G!H is d-semitopological;
(b) hj33NK(G):NK(G)!H is open;
(c) NK(G)\kerh is dense inkerh.
3. Thin sets.
In the introduction we have defined U-thin sets M of a topological groupG, whereUis a neighborhood ofeGinG. The subsetsMofGthat are U-thin for everyUare precisely the thin setsin the sense of Tkachenko [10, 11]. For example compact sets are thin.
The concept ofU-thin subset of a topological groupGis not symmetric and it is possible to give the symmetric definition ofright U-thinset. Note that for every symmetric subset (for example for every subgroup) ofGthe two concepts coincide. MoreoverMis a leftU-thin subset if and only ifM 1 is a rightU-thin subset. Note thatMisU-thin if and only if there exists a neighborhoodU1 ofeGinGsuch thatxU1x 1U for everyx2M.
LEMMA3.1. Let G be a topological group, U;V neighborhoods of eGin G and MG. Then:
(a) if M is U-thin, then every NM is U-thin;
(b) if M1and M2are U-thin, then M1[M2 is U-thin;
(c) if M is U-thin, then Mg and gM are U-thin for every g2G;
(d) if V U and M is V-thin, then M is U-thin;
(e) if M is U-thin and V-thin, then M is U\V-thin;
(f) if f :G!H is a continuous surjective homomorphism and W is a neighborhood of eH in H, then f 1(M) is f 1(W)-thin whenever MH is W-thin.
Consequently, for a topological groupGand a neighborhoodUofeGin G, the family IU(G) fMG:M isU-thin} is a translations invariant
ideal. Observe that Z(G)2 IU(G) for every U2 V(G;t)(eG), that is Z(G)2 I(G)T
fIU(G):U2 V(G;t)(eG)g fMG:Mis thing.
LEMMA3.2. Let G be a topological group, N a normal subgroup of G and q:G!G=N the canonical projection. Moreover let U be a neigh- borhood of eGin G and MG.
(a) If M is U-thin, then q(M)is q(U)-thin.
(b) If M is thin, then q(M)is thin.
DEFINITION 3.3. A topological group (G;t) has small invariant neighborhoods (i.e. G is SIN) if for every U2 V(G;t)(eG) there exists U02 V(G;t)(eG) such thatU0UandgU0g 1U0for everyg2G(i.e.U0is invariant).
Iftis a linear group topology on a groupG, then (G;t) is SIN. A to- pological groupGis SIN if and only ifGis thin. The next lemma shows that for SIN groups the characterization given by Theorem 1.3 is simpler, because condition (a) is always verified since a SIN groupGis thin, and so only condition (b) remains.
LEMMA 3.4. Suppose that the topological group(G;t)is SIN. Then a continuous isomorphism f :(G;t)!(H;s), where(H;s)is a topological group, is semitopological if and only if for every U2 V(G;t)(eG) and for every g2G there exists Vg2 V(H;s)(eH)such that[g;f 1(Vg)]U.
4. Internal approximations for semitopological homomorphisms.
Let f :(G;t)!(H;s) be a continuous surjective homomorphism of topological groups. Inspired by Theorem 1.3 we try to find some condition internal to the groupsGandHto describe whenf is semitopological. The advantage of these internal conditions is obvious. To do this we introduce some concepts which turn out to be equivalent to semitopological for continuous isomorphisms.
First we give some definitions and basic properties.
DEFINITION 4.1. Let (G;t);(H;s) be topological groups and f :(G;t)!(H;s) a continuous surjective homomorphism. Call the topolo- gical group (G;ee t) an A-extension of f :(G;t)!(H;s) i f (G;ee t) contains (G;t) as a normal topological subgroup and there exists an open continuous
homomorphism ef :(G;ee t)!(H;s) extending f. The homomorphism ef is calledassociatedto the A-extension (G;e et).
In the next proposition we give the first properties of the homo- morphism associated to an A-extension of a semitopological homomorph- ism. This proposition generalizes [2, Proposition 3] to semitopological homomorphisms.
PROPOSITION 4.2. Let (G;t);(H;s) be topological groups and f :(G;t)!(H;s) a semitopological homomorphism. If (G;ee t) with fe:(G;ee t)!(H;s)is an A-extension off, then:
(a) G\keref kerf ; (b) GeGkeref ; (c) [keref;G]kerf ;
(d) kerf is a normal subgroup ofG;e
(e) G=kere f is isomorphic to G=kerfkeref=kerf .
PROOF. (a) and (b) are obvious, (c) follows from (a) and (b) because G is a normal subgroup of G, (d) follows from (a) and (e) follows frome
(a), (b) and (d). p
Obviously a continuous surjective homomorphism is semitopological if and only if it has an A-extension.
In the proof of Theorem 1.3 an A-extension of a continuous isomorph- ism of topological groups f :(G;t)!(H;s) is constructed taking GeGHandfW(U;V):U2 V(G;t)(eG);V2 V(H;s)(eH)g, whereW(U;V)
f(uf 1(v);v):u2U;v2Vg, as a base ofV
(eG;et)(eG). Moreoveref :Ge!H is defined by ef(g;h)f(g) for every g2G. We do an analogous con- struction in Definition 4.5 and use it in the proof of Theorem 4.20.
DEFINITION 4.3. Suppose that G, H are groups and f :G!H i s a surjective homomorphism. Asectionis an injective functions:H!Gsuch thatf(s(h))hfor everyh2Hands(eH)eG:
For a surjective group homomorphism there exists always a section.
REMARK 4.4. Let G,Hbe groups andf :G!Ha surjective homo- morphism. Let s:H!G be a section of f. Then Gs(H)kerf and s(H)\kerf feGg. Moreover, if X is a subset of H, then f 1(X)
s(X)kerf.
DEFINITION 4.5. Let (G;t), (H;s) be topological groups, f :(G;t)!(H;s) a continuous surjective homomorphism andsa section of f. Define the filter base
Fs;(t;s) fWs(U;V):U2 V(G;t)(eG);V 2 V(H;s)(eH)g;
where
Ws(U;V) f(us(v);v):u2U;v2Vg:
In caseFs;(t;s)is a base of the neighborhoods of (eG;eH) i nGHof a group topology onGH, then we denote this group topology byas(t;s).
LEMMA 4.6. Let f :(G;t)!(H;s) be a continuous surjective homo- morphism of topological groups, N as-open subgroup of H and s a section of f . IfFs;(t;sj33N)is a base of the neighborhoods of(eG;eH)in GN of a group topologyas(t;sj33N)on GN, then(GN;as(t;sj33N)), withef :GN!H defined byef(g;n)f(g)for every(g;n)2GN, is an A-extension of f . In particular f is semitopological.
PROOF. Consider GN endowed with the group topologyas(t;sj33N).
Theni:G!GN, defined byi(g)(g;eH) for everyg2G, is a topolo- gical embedding; indeed Ws(U;V)\(G feHg)U feHg for every Ws(U;V)2 Fs;(t;sj33
N). LetV 2 V(H;s)(eH). There existsV02 V(N;sj33
N)(eH) such that V0V0V. Since f :(G;t)!(H;s) is continuous, there exists U2 V(G;t)(eG) such that f(U)V0. Then ef(Ws(U;V0))f(U)V0 V0V0V and so ef is continuous. If Ws(U;V)2 Fs;(t;sj33N), then ef(Ws(U;V))f(U)VV and soefis also open. p In the introduction we have used the graph of a surjective homo- morphism of topological groups f :(G;t)!(H;s) to find a topological group (G;ee t) of which (G;t) is a topological subgroup and an open con- tinuous homomorphism ef :(G;e et)!(H;s) extending f: in fact we have takenGe GHwith the product topology andef p2, sinceGf endowed with the topology inherited from GH is topologically isomorphic to (G;t). But the graph is not helpful when we askGto be also normal inG,e because the graph is a normal subgroup ofGHin a very particular case, as we show in the next remark. Anyway it helps in finding a first sufficient condition for a continuous surjective homomorphism to be semitopological (see Corollary 4.8).
REMARK4.7. LetG;Hbe groups andf :G!Ha homomorphism. If
His abelian, thenGf is a normal subgroup ofGH. Iffis surjective, then Gf is a normal subgroup ofGHif and only ifHis abelian.
Theorem 1.1 and Remark 4.7 imply the following corollary that gives a sufficient condition for a continuous surjective group homomorphism to be semitopological.
COROLLARY 4.8. Let G; H be topological groups and f :G!H a continuous surjective homomorphism. If H is abelian, then f is semi- topological.
The converse implication does not hold in general:
EXAMPLE 4.9. Let H be a discrete non-abelian group. Since every continuous surjective homomorphismf :G!H, whereGis a topological group, is open, thenf is semitopological.
In general every non-open continuous surjective homomorphism of topological groups with abelian codomain shows that semitopological does not imply open.
4.1 ±A-open homomorphisms.
Trying to find a characterization of semitopological homomorphisms, we introduce the following concept, which for continuous group iso- morphisms is equivalent to semitopological. In the case of continuous surjective group homomorphisms we prove in Theorem 4.12 that it is a necessary condition.
DEFINITION 4.10. Let (G;t) and (H;s) be topological groups. A homomorphismf :(G;t)!(H;s) i sA-openif for everyU2 V(G;t)(eG)
(a) there existsV2 V(H;s)(eH) such thatf 1(V) i sf 1(f(U))-thin;
(b) for every g2G there exists Vg2 V(H;s)(eH) such that [g;f 1(Vg)]f 1(f(U)).
Every open continuous homomorphismf :(G;t)!(H;s) is A-open. In fact if U2 V(G;t)(eG), since f(U)2 V(H;s)(eH), then: there exists V 2 V(H;s)(eH) such that V3f(U) and so f 1(V) i s f 1(f(U))-thin by Lemma 3.1(f); for every g2G there exists Vg2 V(H;s)(eH) such that
[f(g);Vg]f(U), being f(g)7![f(g);x] (x2H) continuous, and so [g;f 1(Vg)]f 1(f(U)).
REMARK4.11. Theorem 1.3 says thata continuous group isomorph- ism f :(G;t)!(H;s)is semitopological if and only if f is A-open.
In the next theorem we prove that the necessity holds also for continuous surjective group homomorphisms, i.e. that semitopological implies A-open for continuous surjective group homomorphisms. Hence from this theorem we get that open implies A-open, as we have proved directly before.
THEOREM4.12. Let(G;t);(H;s)be topological groups and f :(G;t)!
!(H;s)a continuous surjective homomorphism. If f is semitopological then f is A-open.
PROOF. Let (G;ee t) with ef :Ge !H be an A-extension of f :(G;t)!
!(H;s). LetU2 V(G;t)(eG). There exists a neighborhoodWofeGeeGin (G;ee t) such thatW\GU.
There exists a symmetricW02 V
(eG;et)(eG) such thatW0W0W0W. De- fine ef(W0)V2 V(H;s)(eH) and U0W0\G2 V(G;t)(eG). We prove that f 1(V) i s f 1(f(U))-thin. Equivalently we show that xU0x 1f 1(f(U)) for every x2f 1(V). Let x2f 1(V) and g2U0. Since f 1(V)
f 1(ef(W0))ef 1(ef(W0))W0keref, there existex2Wandeb2kerefsuch thatxexeb. Then
f(xgx 1)f(exebgeb 1ex 1)f(exgex 1)
andexgex 12W\GU. Hencef(xgx 1)2f(U) and soxgx 12f 1(f(U)).
For every g2G there exists a neighborhood W02 V
(eG;et)(eG) of eeG in Ge such that [g;W0]W by the continuity of [g; ]:Ge!G. Defi nee Vgef(W0)2 V(H;s)(eH). We prove that [g;f 1(Vg)]f 1(f(U)). Let x2f 1(Vg); as noted previously there exist ex2W0 and eb2keref such that xexeb. Therefore
f(gxg 1x 1)f(gexebg 1eb 1ex 1)f(gexg 1ex 1);
where gexg 1ex 12W\GU. p
4.2 ±A-open and strongly A-open homomorphisms.
Since we have proved only that for continuous surjective group homo- morphisms A-open is weaker than semitopological and we would like to
have a condition that implies semitopological, in Definition 4.16 we give two properties stronger than A-open.
DEFINITION 4.13. A map t:H!G between topological groups is a quasihomomorphismift(eH)eGand the maps
t0:HH!G, defined byt0(h1;h2)t(h1)t(h2)t(h1h2) 1for every h1;h22H, and
ty:H!G for every y2H, obtained putting ty(h)
t(y)t(h)t(y) 1t(yhy 1) 1for everyh2H, are continuous at (eH;eH) andeH respectively.
This definition is the counterpart of the definition of quasihomo- morphism given in [8] for abelian groups. In fact, ifHandGare abelian,ty
is the identity map for every y2Hand the continuity oft0 at (eH;eH) i s exactly the condition given in [8].
If a mapt:H!Gbetween topological groups is a homomorphism or simplytis continuous ateH, thentis a quasihomomorphism.
REMARK4.14. A map t:(H;iH)!(G;dG) is a quasihomomorphism if and only if it is a homomorphism. In fact, iftis a quasihomomorphism, then t0(H;H) feGgandteH(H) feGg. Thereforetis a homomorphism.
CLAIM4.15. Let f :(G;t)!(H;s)be a continuous surjective homo- morphism of topological groups. Ifkerf Nt, then every section s of f is a quasihomomorphism.
PROOF. For allh1;h22H
f(s0(h1;h2))f(s(h1)s(h2)s(h1h2) 1)f(s(h1))f(s(h2))f(s(h1h2)) 1eH
and sos0(h1;h2)2kerf. Since kerf Ufor eachU2 V(G;t)(eG),s0 is con- tinuous ateH. Analogously lety2H; for everyh2H
f(sy(h))f(s(y)s(h)s(y) 1s(yhy 1) 1)eH
and so sy(h)2kerf. Since kerf U for each U2 V(G;t)(eG), sy is con-
tinuous ateH. p
DEFINITION4.16. Let (G;t) and (H;s) be topological groups. A sur- jective homomorphismf :(G;t)!(H;s) i sA-openif there exists a section soff such that for everyU2 V(G;t)(eG)
(a) there existsV2 V(H;s)(eH) such thats(V) i sU-thin;
(b) for every g2G there exists Vg2 V(H;s)(eH) such that [g;s(Vg)]U.
Ifsis also a quasihomomorphism,f isstrongly A-open.
Note thats0(HH)[S
y2Hsy(H)kerf.
For continuous group isomorphisms the conditions A-open and strongly A-open coincide and they coincide also with A-open. For homo- morphisms only one implication holds:
PROPOSITION 4.17. Let (G;t);(H;s) be topological groups and f :(G;t)!(H;s) a continuous surjective homomorphism. If f is A- open, thenf is A-open.
PROOF. There exists a sectionsoff that witnesses thatf i s A-open.
LetU2 V(G;t)(eG). There existsV2 V(H;s)(eH) such thats(V) i sU-thin. We prove thatf 1(V) i sf 1(f(U))-thin. There existsU02 V(G;t)(eG) such that xU0x 1Ufor everyx2s(V). Thenf 1(f(xU0x 1))f 1(f(U)) for every x2s(V). For every y2f 1(V) there exists x2s(V) such that yU0y 1f 1(f(xU0x 1))f 1(f(x)f(U0)f(x) 1). Indeed, if y2f 1(V), thenf(y)v2V. Putxs(v)2s(V). Thusf(y)f(x) and so, for every u2U0,
f(yuy 1)f(y)f(u)f(y) 1f(x)f(u)f(y)2f(x)f(U0)f(x):
This means thatyuy 12f 1(f(x)f(U0)f(x) 1)f 1(f(xU0x 1)) for every u2U0. HenceyU0y 1f 1(f(U)) for everyy2f 1(V).
Letg2G. There existsVg2 V(H;s)(eH) such that [g;s(Vg)]U. Then f 1([f(g);Vg])f 1([f(g);f(s(Vg))])f 1(f([g;s(Vg)]))f 1(f(U)):
Lety2f 1(Vg). Thenf([g;y])[f(g);f(y)]2[f(g);Vg] and this yields [g;f 1(Vg)]f 1([f(g);Vg])f 1(f(U));
which completes the proof. p
Example 5.21 shows that the converse implication of this proposition does not hold in general. Moreover we don't know whether in general open implies A-open for continuous surjective homomorphisms of topological groups. The question is open in general but we have positive answer in some particular cases: for example when the continuous surjective homo- morphism of topological groups f :(G;t)!(H;s) is an isomorphism or when kerf Nt.
REMARK 4.18. Let f :(G;t)!(H;s) be a continuous surjective homomorphism. If kerf Nt, i.e. kerf Ufor everyU2 V(G;t)(eG), then t is the initial topology of tq on G=kerf. In this situation it is possible to think that U2 V(G;t)(eG) is such that Uq 1(q(U)), where q:G!G=kerf is the canonical projection. Continuous isomorphisms trivially satisfy this condition.
Letf :G!Hbe a surjective homomorphism. There exists a sections of f which is a homomorphism if and only if G is isomorphic to the semidirect product kerfj H. In this case, when the topology on Gkerfj His the product topology, strongly A-open obviously coin- cides with A-open.
PROPOSITION 4.19. For a continuous surjective homomorphism of topological groups f:(G;t)!(H;s), in case (G;t)(kerfj H;tj33kerf tj33H), iff is open (i.e.tj33Hs) thenf is strongly A-open (i.e. A-open).
PROOF. Let Lkerf and suppose without loss of generality that GLj H; thenf p2ands:H!G, defined bys(h)(eL;h) for every h2H, is a section off and a homomorphism. So it suffices to verify thats makesf A-open. LetU2 V(G;t)(eG). There existsU0WV2 V(G;t)(eG), where W2 V(L;tj33L)(eL) and V 2 V(H;s)(eH), such that U0U0U0U. Si nce s(V)U0, thens(v)U0s(v) 1Ufor everyv2V, that iss(V) i sU-thin. Let g2G. There exists U02 V(G;t)(eG) such that U0U0U and there exists UgWgVg2 V(G;t)(eG), whereWg2 V(L;tj33
L)(eL) andVg2 V(H;s)(eH), such thatUgU0andgUgg 1U0. Letv2Vg. Then, sinces(Vg)Ug,
[g;s(v)]gs(v)g 1s(v) 12gs(Vg)g 1s(Vg)gUgg 1UgU0U0U:
Hence [g;s(Vg)]U and this completes the proof thatf is strongly A-
open. p
Example 5.6 shows that the existence of a section which is a quasiho- momorphism of a continuous surjective group homomorphism f does not yield in general thatf i s A-open.
Theorem 4.20 is one of the main results of this paper. The technique used in its proof is inspired by that of the proof in [2] of Theorem 1.3.
THEOREM 4.20. Let (G;t);(H;s) be topological groups, N a s-open subgroup of H, f :(G;t)!(H;s)a continuous surjective homomorphism and s a section of fj33f 1(N):f 1(N)!N. Then Fs;(t;sj33
N) is a base of the
neighborhoods of(eG;eH)in GN of a group topologyas(t;sj33N)on GN if and only if s makes f strongly A-open.
PROOF. Suppose that Fs;(t;sj33
N) (see Definition 4.5) is a base of the neighborhoods of (eG;eH) i nGNof a group topologyas(t;sj33N) onGN and let U2 V(G;t)(eG). So W(U;V)Ws(U;V)2 Fs;(t;sj33
N) for some V 2 V(N;sj33
N)(eH). Then there exists W(U0;V0)2 Fs;(t;sj33
N) such that W(U0;V0)W(U0;V0)W(U;V). Therefore (us(v);v)(u0s(v0);v0)2W(U;V) for everyu;u02U0;v;v02V0and this yields
(us(v)u0s(v0);vv0)(us(v)u0s(v0)s(vv0) 1s(vv0);vv0)2W(U;V):
Choosinguu0eGwe gets(v)s(v0)s(vv0) 12Ufor every (v;v0)2V0V0 and sos0is continuous at (eH;eH).
Let y2H, x2G and W(U;V)2 Fs;(t;sj33
N). Then there exists W(U0;V0)2 Fs;(t;sj33
N) such that
(x;y)W(U0;V0)(x;y) 1W(U;V):
Hence (x;y)(us(v);v)(x 1;y 1)2W(U;V) for everyu2U0;v2V0and so W(U;V)3(xus(v)x 1;yvy 1)(xus(v)x 1s(yvy 1) 1s(yvy 1);yvy 1):
Takingxs(y) andueGwe get in particulars(y)s(v)s(y) 1s(yvy 1) 12 2Ufor everyv2V0and hencesyis continuous ineH.
This proves thatsis a quasihomomorphism.
To prove that s makesf strongly A-open it remains to prove that s makesf A-open. LetU2 V(G;t)(eG) andV 2 V(H;s)(eH). Then there exists U02 V(G;t)(eG) such thatU0is symmetric andU0U0U. Let
s00:H!Gdefined bys00(h)s0(h;h 1)s(h)s(h 1) for everyh2H:
The continuity of s0 at (eH;eH) implies the continuity ofs00ateHbecause s(eH)eG. Since s00 is continuous at eH, so there exists a symmetric V02 V(H;s)(eH) such that s00(V0)U. By the hypothesis there exist U002 V(G;t)(eG) and a symmetric V002 V(H;s)(eH) such that V00V0 and W(U00;V00) 1W(U0;V). Hence (us(v);v) 1(s(v) 1u 1;v 1)2W(U0;V) for everyu2U00;v2V00. Consequentlys(v) 1u 1s(v 1) 12U0. SinceU0is symmetrics(v 1)us(v)2U0for everyu2U00;v2V00. Moreovers(v)us(v 1)2 2U0 for every u2U00;v2V00, because V00 is symmetric. Since s00(V00) s00(V0)U0, for everyv2V00there existsu02U0such thats(v 1)s(v) 1u0. Then for everyu2U00,v2V00there existsu02U0such that
s(v)us(v) 1s(v)us(v 1)u02U0U0U:
This means thats(V00) i sU-thin.
Let g2G. By the hypothesis there exist U02 V(G;t)(eG) and V02 V(H;s)(eH) such that
(g;eH)W(U0;V0)(g 1;eH)W(U;V):
Consequently (gus(v)g 1;v)2W(U;V) for every u2U0;v2V0. Then gus(v)g 1s(v) 12Ufor everyu2U0;v2V0. TakeueG; so [g;s(v)]2U for everyv2V0, that is [g;s(V0)]U. Hencef is strongly A-open.
Suppose thats0andsy(for everyy2H) are continuous at (eH;eH) and eHrespectively. We want to prove that the filter baseFs;(t;sj33
N) is a base of the neighborhoods of (eG;eH) i nGN. LetW(U;V)2 Fs;(t;sj33N).
(1) There exist U02 V(G;t)(eG) and V02 V(N;sj33N)(eH) such that U0U0U0U and V0V0V. Since fj33f 1(N) :f 1(N)!N i s A-open, there existsV002 V(N;sj33N)(eH) such thatV00V0ands(V00) i sU0-thin; in particular there existsU002 V(G;t)(eG) such thatU00U0andxU00x 12U0for every x2s(V00). Thanks to the continuity of s0 at (eH;eH) there exists V0002 V(N;sj33
N)(eH) such that V000V00 and s(v)s(v0)2U00s(vv0) for every v;v02V000. Then W(U00;V000)W(U00;V000)W(U;V): for every u;u02U00 andv;v02V000 there existsu002U00such that
(us(v);v)(u0s(v0);v0)(us(v)u0s(v0);vv0)(u(s(v)u0s(v) 1)s(v)s(v0);vv0)
(u(s(v)u0s(v) 1)u00s(vv0);vv0)2W(U;V):
(2) There exists U02 V(G;t)(eG) such that U0U0U and U0 is sym- metric. As shown before, the continuity of s0 at (eH;eH) implies the con- tinuity ateHof the previously defineds00. Sinces00is continuous ateH, there existsV02 V(N;sj33
N)(eH) such thatV0Vands(v)s(v 1)2U0for everyv2V0. Sincefj33f 1(N):f 1(N)!N i s A-open, there exists V00 2 V(N;sj33
N)(eH) such thatV00is symmetric,V00V0ands(V00) i sU0-thin; in particular there exists U002 V(G;t)(eG) such that U00 is symmetric and xU00x 1U0 for every x2s(V00). Then W(U00;V00) 1W(U;V). In fact for every u2U00 and v2V00there existsu02U0such that
(us(v);v) 1(s(v) 1u 1;v 1)(s(v) 1u 1s(v 1) 1s(v 1);v 1)
((s(v) 1u 1s(v))u0s(v 1);v 1)2W(U;V):
(3) Let (x;y)2GN. There existU02 V(G;t)(eG) andV02 V(N;sj33N)(eH) such thatU0 is symmetric,U0U0U0U0U andyV0y 1V. There exists also U002 V(G;t)(eG) such thatxU00x 1U0 and s(y)U00s(y) 1U0. Si nce fj33f 1(N):f 1(N)!Ni s A-open, there exists a symmetricV002 V(N;sj33
N)(eH)
such that V00V0, [x;s(V00)]U0 and [s(y) 1;s(V00)]U00. By the con- tinuity of sy ateH there existsV0002 V(N;sj33N)(eH) such that V000 V00 and s(y)s(v)s(y) 12U0s(yvy 1) for everyv2V000. Then (x;y)W(U0;V00)(x;y) 1 W(U;V): for everyu2U00andv2V000 there existsu02U0such that (x;y)(us(v);v)(x 1;y 1)(xus(v)x 1;yvy 1)
((xux 1)(xs(v)x 1)s(yvy 1) 1s(yvy 1);yvy 1)2W(U;V) becauseyvy 12yV0y 1V and
(xux 1)(xs(v)x 1)s(yvy 1) 1(xux 1)(xs(v)x 1)s(y)s(v) 1s(y) 1u0
(xux 1)(xs(v)x 1s(v) 1)s(y)(s(y) 1s(v)s(y)s(v) 1)s(y) 1u02U:
This completes the proof. p
The next corollary is one of the most interesting results of this paper because it gives a sufficient condition for a continuous surjective homo- morphism to be semitopological:
COROLLARY 4.21. Let f :(G;t)!(H;s) be a continuous surjective homomorphism. Iff is strongly A-open, thenf is semitopological.
PROOF. There exists a sectionsoffthat makesfstrongly A-open. By Theorem 4.20Fs;(t;s)is a base of the neighborhoods of (eG;eH) i nGHof a group topologyas(t;s). Thenf is semitopological by Lemma 4.6. p The converse implication does not hold in general Ð see Example 5.13.
We have also the following corollary of Theorem 4.20.
COROLLARY4.22. Let (G;t);(H;s)be topological groups,N as-open subgroup of H and f :(G;t)!(H;s) a continuous surjective homo- morphism such that fj33f 1(N):f 1(N)!N is A-open. If there exists a section s of f which is a homomorphism, then f:(G;t)!(H;s) is semitopological.
As a consequence of Proposition 4.19 and Corollary 4.22 we have that for a continuous surjective homomorphism of topological groups f :(G;t)!(H;s), ifGkerfj Handttj33kerf tj33H, in particular there exists a sectionsoffwhich is a homomorphism and thenf open)smakes f A-open)f semitopological.
Example 5.6 shows thatshomomorphism does not imply thatsmakesf A-open.
REMARK 4.23. Corollary 4.21 together with Theorem 4.12 imply Theorem 1.3. Indeed letf :(G;t)!(H;s) be a continuous isomorphism of topological groups; Theorem 1.3 states thatfis semitopological if and only if f is A-open. Sincef 1 is a homomorphism, in particular it is a quasihomomorphism. Since forf continuous isomorphism A-open is the same as A-open as noted before, Corollary 4.21 and Theorem 4.12 prove Theorem 1.3.
4.3 ±Strongly A-open homomorphisms.
In this section we introduce another property stronger than A-open:
DEFINITION 4.24. Let (G;t) and (H;s) be topological groups. A homomorphism f :(G;t)!(H;s) i s strongly A-open if for every U2 V(G;t)(eG)
(a) there existsV2 V(H;s)(eH) such thatf 1(V) i sU-thin;
(b) for every g2G there exists Vg2 V(H;s)(eH) such that [g;f 1(Vg)]U.
From the definition we have directly that if f :(G;t)!(H;s) i s a strongly A-open continuous surjective group homomorphism, then
[G;kerf]Nt:
In particular kerf isU-thin for everyU2 V(G;t)(eG).
Every strongly A-open homomorphism is A-open and so also A-open by Proposition 4.17. Note that Example 5.6 shows that the existence of a section which is a quasihomomorphism does not imply the condition strongly A-open.
LEMMA 4.25. Let (G;t);(H;s) be topological groups and f :(G;t)!
!(H;s) a continuous surjective homomorphism. Suppose that t is Hausdorff and Z(G) feGg. If f is strongly A-open, then f is a (semi- topological) isomorphism.
PROOF. Si nce t is Hausdorff, the previous observation implies that kerfZ(G). SinceZ(G) feGg, kerf is trivial andf is an isomorphism. p
The previous lemma yields that in general semitopological cannot imply strongly A-open.
Moreover we construct the following example, which shows that open together with (strongly) A-open do not imply strongly A-open. Conse- quently strongly A-open cannot be equivalent to semitopological in gen- eral. However we see in §6 that the class of all strongly A-open continuous surjective homomorphisms has good categorical properties. This allows us to find some nice property of the classSi.
EXAMPLE4.26. LetKbe a compact Hausdorff simple group (take for exampleKSO(3;R)). LetGKKandf p1:KK!K. First of allf is open. Moreoverf is not strongly A-open: if it was strongly A-open, sinceZ(K) is trivial, then it would be an isomorphism by Lemma 4.25, but kerf feKg K. The sections:K!Goff defined bys(x)(x;eK) for everyx2Kis a homomorphism. Hencef is strongly A-open by Proposi- tion 4.19.
REMARK 4.27. For a continuous isomorphism of topological groups f :(G;t)!(H;s) it is equivalent to be semitopological, A-open, A-open, strongly A-open and strongly A-open.
EXAMPLE4.28. In view of Remark 4.27 and Corollary 4.8 it suffices to consider a non-open continuous isomorphism of topological abelian groups to see that in general (semitopological and) strongly A-open does not imply open.
The next theorem shows that for a continuous surjective homo- morphims such that its kernel is contained in every neighborhood of the neutral element the four new conditions introduced are equivalent to semitopological. Theorem 1.3 is a corollary of the following theorem, since every continuous isomorphism trivially satisfies the assumption, as noted previously in Remark 4.18. As a consequence of Corollary 6.5, in Corollary 6.6 we add another equivalent condition to those of the next theorem.
THEOREM 4.29. Let (G;t);(H;s) be topological groups and f :(G;t)!(H;s) a continuous surjective homomorphism such that kerf Nt. The following are equivalent:
(a) f is strongly A-open;
(b) f is strongly A-open;
(c) f is A-open;
(d) f is A-open;
(e) f is semitopological.
PROOF. (a))(c) is trivial and (c))(d) by Proposition 4.17.
(d))(a) Let U02 V(G;t)(eG). Then there exists U2 V(G;t)(eG) such that UUU0. Letg2G. Si ncef is A-open there existV;Vg2 V(H;s)(eH) such that f 1(V) i s f 1(f(U))-thin and [g;f 1(Vg)]f 1(f(U)). But f 1(f(U))Ukerf UUU0 and so f 1(V) i s U0-thin and [g;f 1(Vg)]U0.
(e))(d) by Theorem 4.12 and (c))(e) by Claim 4.15 and Corollary 4.21.
(b),(c) by Claim 4.15. p
In the next lemma we consider the case when the topology on the do- main in SIN. In this case the first condition in the definitions of strongly (A-open) and A-open is automatically satisfied.
LEMMA 4.30. Suppose that the topological group(G;t)is SIN and let f :(G;t)!(H;s)be a continuous surjective group homomorphism. Then:
(a) f is A-open if and only if for every U2 V(G;t)(eG)and for every g2G there exists Vg2 V(H;s)(eH) such that [g;f 1(Vg)]
f 1(f(U));
(b) f is A-open if and only if there exists a section s of f such that for every U2 V(G;t)(eG) and for every g2G there exists Vg2 V(H;s)(eH)such that[g;s(Vg)]U;
(c) f is strongly A-open if and only if for every U2 V(G;t)(eG)and for every g2G there exists Vg2 V(H;s)(eH) such that [g;f 1(Vg)]U.
PROPOSITION4.31. Let(G;t)(L;r)(H;s0)(trs0) be a topolo- gical group andf p2:(G;t)!(H;s)be the canonical projection, where sis a group topology onHsuch thatss0. Then the following conditions are equivalent:
(a) f is A-open;
(b) f is A-open;
(c) f is strongly A-open.
Moreover we have the following characterizations:
(i) f is A-open if and only if 1H:(H;s0)!(H;s) is semi- topological;
(ii) f is strongly A-open if and only if L0Nr and1H:(H;s0)!
!(H;s)is semitopological.
