topology proceedings 37 (2011) pp. 33-60: completeness...

Volume 37, 2011

Pages 33–60

http://topology.auburn.edu/tp/

Completeness type properties of

semitopological groups, and thetheorems of Montgomery and Ellis

by

Alexander V. Arhangel’skii and Mitrofan M. Choban

Electronically published on April 29, 2010

Topology Proceedings

Web: http://topology.auburn.edu/tp/Mail: Topology Proceedings

Department of Mathematics & StatisticsAuburn University, Alabama 36849, USA

E-mail: [email protected]: 0146-4124

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TOPOLOGYPROCEEDINGSVolume 37 (2011)Pages 33-60

http://topology.auburn.edu/tp/

E-Published on April 29, 2010

COMPLETENESS TYPE PROPERTIES OF

SEMITOPOLOGICAL GROUPS, AND THE

THEOREMS OF MONTGOMERY AND ELLIS

ALEXANDER V. ARHANGEL’SKII AND MITROFAN M. CHOBAN

Abstract. In this paper the class of fan-complete spaces isintroduced. Every Gδ-subspace of a pseudocompact space isfan-complete. We prove that a paratopological group is atopological group provided it contains a dense fan-completesubspace. Moreover, if a semitopological group contains adense Cech-complete subspace, then it is a Cech-completetopological group. This improves some results of A. Bouziad,P. Kenderov, I.S. Kortezov, and W.B. Moors. Some other newresults are obtained (see, in particular, Theorems 5.1 and 5.3,and Corollaries 5.4, 6.5 and 6.6).

1. Introduction

By a space we understand a regular topological T1-space. We usethe terminology from [7], [17]. Let ω be the first infinite ordinal andthe first infinite cardinal, ω = {0, 1, 2, ...}. By clXH we denote theclosure of a set H in a space X. A paratopological group is a groupwith a topology such that the multiplication is jointly continuous,and a semitopological group is a group with a topology such thatthe multiplication is separately continuous. Every paratopologicalgroup is a semitopological group. A semitopological group in whichthe inverse operation is continuous is called a quasitopological group.

2010 Mathematics Subject Classification. 54H11, 54H20, 54H15.Key words and phrases. Topological group, semitopological group, paratopo-

logical group, Baire property, fan-complete space, Cech-complete space, analyticset, pseudocompact space.

c⃝2010 Topology Proceedings.

33

34 ALEXANDER V. ARHANGEL’SKII AND MITROFAN M. CHOBAN

The space S of reals with the topology generated by the baseconsisting of the sets [a, b) = {x ∈ R : a ≤ x < b}, where a, b ∈R and a < b, is called Sorgenfrey line. Sorgenfrey line has thefollowing properties (see [7]):

- S is an Abelian paratopological group with the Baire property;- S is a hereditarily Lindelof first-countable hereditarily separable

non-metrizable space;- Each metrizable subspace of S is countable.- Each compact subspace of S is countable.- S does not admit a structure of a topological group.In 1936 D. Motgomery [28] has proved the following two theo-

rems:Theorem 1M. Every separable semitopological group metrizableby a complete metric is a topological group.Theorem 2M. Every metrizable by a complete metric semitopo-logical group is a paratopological group.

In 1957 R. Ellis [16] showed that every locally compact semi-topological group is a topological group. Further interesting resultson semitopological and paratopological groups were established byZ. Zelazko [38], N. Brand [11], L. G. Brown [12], P. Kenderov, I.S. Kortezov, and W. B. Moors [23], E. A. Reznichenko [34], andother authors (see [7], [20], [25], [33]). Some important recent ad-vances in this direction were made by A. Bouziad (see [9], [10]).In [10] A. Bouziad has proved that every Cech-complete (everyCech-analytic with the Baire property) semitopological group is atopological group. A. V. Arhangel’skii and E. A. Reznichenko (see[7], [8]) have proved that a paratopological group G is a topologicalgroup provided it is a Gδ-subspace of some pseudocompact space.In [23] P. Kenderov, I. S. Kortezov and W. B. Moors introduced theclass of strongly Baire (strongly β-unfavorable) spaces and provedthat a strongly Baire semitopological group is a topological group.Some remarkable relations between separate continuity and jointcontinuity were established in [25], [30], [31], [37], [19], [34]. Themonograph [7] contains many references of this kind.

In this paper we develop the methods from [7] and extend thetheorems of D. Montgomery and R. Ellis over the very wide classof fan-complete spaces (see Theorems 5.1 and 5.2). Fan-completespaces have the following properties:

TOPOLOGY PROCEEDINGS 35

- All compact spaces, all countably compact spaces, and all pseu-docompact spaces are fan-complete.

- Every dense Gδ-subspace of a fan-complete space is fan-com-plete.

- Any image of a fan-complete space under an open continuousmapping is fan-complete.

- Any locally fan-complete space is fan-complete.Besides, the definition of fan-complete spaces is very transparent

and easy to work with.

2. Various types of completeness of topological spaces

For a sequence {Hn :n∈ω} of subsets of a space X, Lim{Hn :n∈ω}is the set of all accumulation points of {Hn : n ∈ ω}. If Hn+1 ⊆ Hn

for any n ∈ ω, then Lim{Hn : n ∈ ω} = ∩{clXHn : n ∈ ω}.A sequence {Un : n ∈ ω} of open subsets of a space X is called

a stable sequence if it satisfies the following conditions:(S1) ∅ = Un+1 ⊆ Un for any n ∈ ω;(S2) Every sequence {Vn : n ∈ ω} of open non-empty sets in X

such that Vn ⊆ Un for each n ∈ ω, has an accumulation point inX, i.e. Lim{Vn : n ∈ ω} = ∅.

A subset L of a space X is bounded if for every locally finitefamily γ of open subsets in X the set {U ∈ γ : U ∩L = ∅} is finite.

A space X is feebly compact if every locally finite family of opensubsets in X is finite, i.e. X is bounded in X. For Tychonoffspaces, feeble compactness is equivalent to pseudocompactness. Ev-ery countably compact space is feebly compact. A subset L of aTychonoff space X is bounded if and only if every continuous func-tion on X is bounded on L (see [7], [17]).

From conditions (S1) and (S2) it follows thatH=∩{clXUn :n∈ω}= Lim{Un : n ∈ ω} is a bounded non-empty subset of X.

A space X is called µ-complete if the closure of each boundedsubset is compact.

Let Y be a dense subspace of a space X, γ={γn={Uα:α∈An} :n ∈ ω} be a sequence of families of open subsets of X, and let π= {πn : An+1 → An : n ∈ ω} be a sequence of mappings. Asequence α = {αn : n ∈ ω} is called a c-sequence if αn ∈ An andπn(αn+1) = αn for every n. Let H(α) = ∩{Uαn ;n ∈ ω}. Considerthe following conditions:


(SC1) ∪{Uβ : β ∈ An} is a dense subset of X for each n ∈ ω.(SC2) ∪{Uβ : β ∈ π−1

n (α)} is a dense subset of the set Uα for allα ∈ An and n ∈ ω.

(SC3) Uα = ∪{Uβ : β ∈ π−1n (α)} for all α ∈ An and n ∈ ω.

(SC4) ∪{clXUβ : β ∈ π−1n (α)} ⊆ Uα for all α ∈ An and n ∈ ω.

(C1) For any c-sequence α = {αn ∈ An : n ∈ ω}, the sequence{Uαn ;n ∈ ω} is stable.

(C2) For any c-sequence α = {αn ∈ An : n ∈ ω}, each sequence{yn ∈ Y ∩ Uαn ;n ∈ ω} has an accumulation point in X.

(C3) For any c-sequence α = {αn ∈ An : n ∈ ω}, the sequence{Uαn ;n ∈ ω} is a base of open neighbourhoods of the set H(α) inX.

(C4) For any c-sequence α = {αn ∈ An : n ∈ ω}, the set H(α) isa non-empty compact subset of X.

(C5) For any c-sequence α = {αn ∈ An : n ∈ ω}, the set H(α) isa non-empty countably compact subset of X.

Sequence γ and π are called a wA-sieve if they have the Property(SC3) and each γn is a cover of X;

They are an A-sieve if they have the Properties (SC3), (SC4)and each γn is a cover of X;

They are called a dense wA-sieve if they have the Properties(SC1), (SC2), and a dense A-sieve if they have the Properties (SC1),(SC2), (SC4).

A space X is called densely sieve-complete if there exist a densesubspace Y and a dense A-sieve with the Properties (C2) and (C4).A space X is called sieve-complete if there exists an A-sieve withthe Properties (C2) and (C4) for Y = X.

A space X is called densely q-complete if there exist a densesubspace Y and a dense A-sieve with the Property (C2). A spaceX is called q-complete if there exists an A-sieve with the Propertiies(C2) and (C5) for Y = X.

A space X is called densely fan-complete if there exists a denseA-sieve on X with the Property (C1). A space X is called fan-complete if there exists an A-sieve on X with the Property (C1).

The sieve-complete and q-complete spaces were examined in [13],[14], [35], [36], [32].

Following E. Michael [27] and [23], a point x ∈ X is called aqD-point if there exist a dense subspace Y of X and a sequence ofneighbourhoods {Un : n ∈ ω} of the point x in X such that if xn ∈Y ∩ Un, then the sequence {xn : n ∈ ω} has a cluster point in X.


If Y = X, then x is a q-point. A space X is called a densely q-spaceif each x ∈ X is a qD-point. A space X is called a q-space if eachx ∈ X is a q-point. Any q-complete space is a q-space.

Let {γn = {Uα : α ∈ An}, πn : An+1 → An : n ∈ ω} be a denselywA-sieve with the Property (C2) and X1 = ∪{H(α) : α is a c-sequence}. Then every point x ∈ X1 is a qD-point. The set X1 isdense in X.

A point x ∈ X is called a point of countable type if there existsa compact subset F with a countable base of open neighbourhoods{Un : n ∈ ω} in X such that x ∈ F . A space X is called a space ofpointwise countable type if each x ∈ X is a point of countable type[3], [17].

Remark 2.1. By Proposition 2.10 from [14], for every wA-sieve U= {γn = {Uα : α ∈ An}, πn : An+1 → An : n ∈ ω} there exist anA-sieve V = {ξn = {Vβ : β ∈ Bn}, qn : Bn+1 → Bn : n ∈ ω} andthe mappings {hn : Bn → An : n ∈ ω} such that:

1. clXVβ ⊆ Uqn(β) and hn ◦ qn = πn ◦ hn+1 for each β ∈ Bn andfor any n ∈ ω.

2. If U has Property (C1), then V has Property (C1) too.3. If U has Properties (C2) and (C3), then V has Properties (C2)

and (C3) too.4. If U has Properties (C2) and (C4), then V has Properties (C2)

and (C4) too.Moreover, if Y is a dense subspace of X and U is a dense wA-

sieve, then V is a dense A-sieve with properties:- the family ξn is disjoint for any n ∈ ω;- q−1

n (β) = {µ ∈ Bn+1 : Vµ ⊆ Vβ} for each β ∈ Bn and for anyn ∈ ω;

- if k ∈ {1, 2, 3, 4, 5} and U has Property (Ck), then V has Prop-erty (Ck), too.

Remark 2.2. Every feebly compact space is fan-complete. Sinceevery Tychonoff space is a closed subspace of some pseudocompactspace, a closed subspace of a fan-complete space needn’t be fan-complete. The following statements are obvious:

- Any closed subspace of a sieve-complete space is sieve-complete;- Any closed subspace of a q-complete space is q-complete;- Any q-complete space is densely q-complete;


- Every sieve-complete space is q-complete, and every q-completespace is fan-complete;

- Every fan-complete µ-complete space is sieve-complete;- A space X is densely fan-complete if and only if it contains a

dense fan-complete subspace.

Proposition 2.3. Every Gδ-subspace of a fan-complete space isfan-complete. In particular, every Cech-complete space is sieve-complete.

Proof. Let U = {γn = {Uα : α ∈ An}, πn : An+1 → An : n ∈ ω}be an A-sieve with the Property (C1) on a space X, and let Y =∩{Un : n ∈ ω}, where {Un : n ∈ ω} is a sequence of open subsetsof X. Then there exist a sequence {ξn = {Vβ : β ∈ Bn} : n ∈ ω} offamilies of open subsets of X, sequences of mappings {qn : Bn+1 →Bn : n ∈ ω} and {hn : Bn → An : n ∈ ω} such that:

- clXVβ ⊆ Un ∩ Uqn(β) and hn ◦ qn = πn ◦ hn+1 for each β ∈ Bn

and for any n ∈ ω;- Vµ∩Y = ∪{Vβ ∩Y : β ∈ q−1

n (µ)} = ∪{Y ∩clXUβ : β ∈ π−1n (α)}

for all µ ∈ Bn and n ∈ ω.Then W = {Wβ = Y ∩ Vβ : β ∈ Bn}, qn : Bn+1 → Bn : n ∈ ω} is

an A-sieve with the Property (C1) on Y . �The next statement can be proved in a standard way.

Proposition 2.4. If a space X contains a dense fan-complete sub-space, then X has the Baire property.

A space X is said to be a complete M -space (see [29]) if thereexists a continuous closed mapping f : X → Y onto a completemetrizable space Y such that the fibers f−1(y) are countably com-pact. In this case we say that f is a quasiperfect mapping. Everycomplete M -space is q-complete.

For a metric space X the following theorem was proved in [15].

Theorem 2.5. Let f : X → Y be an open continuous mapping of aspace X onto a space Y , and X1 be a dense fan-complete subspaceof X. Then there exists a dense fan-complete subspace Y1 of Ysuch that f(X1) ⊆ Y1. Moreover, if X, Y are Tychonoff spaces,and βf : βX → βY is the continuous extension of f , then thereexists a paracompact Gδ-subspace Z of βX such that:

- S = βf(Z) is a paracompact dense subspace of βY ;


- g = βf |Z : Z → S is a perfect mapping, S1 = Y ∩ S is adense Gδ-subspace of Y1 and Z ∩ f−1(y) = X ∩ g−1(y) ⊆ g−1(y) =clβX(Z ∩ f−1(y)) for each y ∈ S1;

- if X1 is a q-complete space, Z1 = Z ∩X and S1 = f(Z1), thenh = f |Z1 : Z1 → S1 is a quasiperfect mapping, and Z1 is a completeM -space;

- if X1 is a sieve-complete space, then Z ⊆ X;- if X or X1 is µ-complete, then Z ⊆ X.

Proof. There exist a sequence γn = {Uα : α ∈ An} : n ∈ ω} offamilies of open subsets of X and a sequence {πn : An+1 → An :n ∈ ω} of mappings such that:

(RC1) X1 ⊆ ∪{Uα : α ∈ A0} and Uα ∩ X1 ⊆ ∪{Uβ : β ∈π−1n (α)} ⊆ Uα for all α ∈ An and n ∈ ω;(RC2) for any c-sequence α = {αn ∈ An : n ∈ ω}, the sequence

{X1 ∩ Uαn ;n ∈ ω} is stable in X1.Let ηn = {Vα = f(Uα) : α ∈ An} for any n ∈ ω, X(α) =

∩{Uαn : n ∈ ω} and Y (α) = ∩{Vαn : n ∈ ω} for any c-sequenceα = {αn : n ∈ ω}. We put X2 = ∪{X(α) : α is a c-sequence} andY1 = ∪{Y (α) : α is a c-sequence}. By the construction, f(X(α)) ⊆Y (α), X1 ⊆ X2 and f(X2) ⊆ Y1.

It is obvious that Y1 ⊆ ∪{Vα : α ∈ A0} and Vα∩Y1 ⊆ ∪{Vβ : β ∈π−1n (α)} ⊆ Vα for all α ∈ An and n ∈ ω. Let α = {αn : n ∈ ω} be ac-sequence and {Vn : n ∈ ω} be a sequence of open non-empty setsin Y such that Vn ⊆ Vαn for any n ∈ ω. We put Un = Uαn∩f−1(Vn).Then Lim{Un : n ∈ ω} = ∅ and f(Lim{Un : n ∈ ω}) ⊆ Lim{Vn :n ∈ ω}. Thus, Y1 is a dense fan-complete subspace of Y .

Assume that X and Y are Tychonoff spaces. Then there aredisjoint families {ξn = {Hβ : β ∈ Bn} : n ∈ ω} of open subsetsof βX, disjoint families {ξn = {Wβ : β ∈ Bn} : n ∈ ω} of opensubsets of βY , mappings {qn : Bn → An : n ∈ ω} and mappings{pn : Bn+1 → Bn : n ∈ ω} such that:

- Wn = {Wβ : β ∈ Bn} is an open dense subset of βY for eachn ∈ ω};

- ∪{clβXHµ : µ ∈ p−1n (β)} ⊆ Hβ and ∪{clβYWµ : µ ∈ p−1

n (β)} ⊆Wβ for all β ∈ Bn and n ∈ ω};

- X ∩Hβ ⊆ Upn(β) and βf(Hβ) =Wβ for all β ∈ Bn and n ∈ ω};- πn ◦ qn+1 = qn ◦ pn+1 for each n ∈ ω}.


We put Hn = {Hβ : β ∈ Bn}, Z = ∩{Hn : n ∈ ω} and S =∩{Wn : n ∈ ω}. The subspaces Z, S and the mapping g = βf |Z :Z → S are what we want. �Corollary 2.6. If a µ-complete space X contains a dense fan-complete subspace, then X contains a dense Cech-complete para-compact subspace.

Theorem 2.7. Every paracompact fan-complete space is Cech-com-plete.

Proof. If X is a paracompact fan-complete space, then there exista sequence {γn = {Uα : α ∈ An} : n ∈ ω} of open locally finitecovers and a sequence {πn : An+1 → An : n ∈ ω} of mappingswith the properties (C1), (C2) such that Uα is a Fσ-set in X for allα ∈ An and n ∈ ω. Fix continuous non-negative functions {gα : α ∈An, n ∈ ω} such that Σ{gβ : β ∈ An} = 2−n and g−1

α (0) = X \ Uα

for all α ∈ An and n ∈ ω. Then ρ(x, y) = Σ{gα : α ∈ An, n ∈ ω}is a continuous pseudometric on X. Thus, there exist a metricspace (Y, d) and a continuous mapping f : X → Y onto Y suchthat ρ(x, y) = d(f(x), f(y)) for all x, y ∈ X. Fix b ∈ X and ac-sequence α = {αn : n ∈ ω} such that b ∈ ∩{Uαn : n ∈ ω}. Thereexists a sequence {ε(n) : n ∈ ω} such that ε(n) > ε(n + 1) > 0and Vn = {y ∈ X : ρ(x, y) < ε(n)} ⊆ Uαn for any n ∈ ω. Sincef−1(f(b)) ⊆ H = ∩{Uαn : n ∈ ω} and H is a bounded subset of theparacompact space X, the sets f−1(f(b)) and clXH are compact,and f is a perfect mapping. Now it is easy to verify that (Y, d) isa complete metric space. �

The following facts are easy to establish:

Proposition 2.8. Let X be a space such that for every non-emptysubset U there exist an open non-empty subset V and a fan-completesubspace Z of X such that Z ⊆ V ⊆ U ⊆ clXZ. Then X containsa dense fan-complete subspace.

Proposition 2.9. Any locally fan-complete space is fan-complete.

Proposition 2.10. Let G be a semitopological group or a homo-geneous space, V be an open subset of G, Z be a fan-complete sub-space, and let Z ⊆ V ⊆ clXZ. Then:

1. G contains some dense fan-complete subspace.2. If Z = V , then G is fan-complete.


A set Z is normally in a space X if for any closed subset F ofthe subspace Z and each open subset U of X that contains F thereexists an open subset V of X such that F ⊆ V ⊆ clXV ⊆ U .Obviously, that Z is a normal closed subspace of X.

Proposition 2.11. Let Y be a dense subset of X and {Un : n ∈ ω}be a sequence of open non-empty subsets of X with the properties:

- F = ∩{Un : n ∈ ω} is normally in X subset;- clXUn+1 ⊆ Un for each n ∈ ω;- each sequence {yn ∈ Y ∩Un : n ∈ ω} has an accumulation point

in X.Then:1. F is a closed normal subspace of the space X.2. The subspace F is countably compact.3. {Un : n ∈ ω} is a base of the set F in X.

Proof. Assertion 1 is obvious. Let L = {xn ∈ F : n ∈ ω} be adiscrete sequence of the subspace F . Then there exists a disjointfamily {Vn : n ∈ ω} of open subsets of X such that xn ∈ Vnfor any n ∈ ω. There exists an open subset W of X such thatL ⊆ W ⊆ clXW ⊆ ∪{Vn : n ∈ ω}. We put Wn = W ∩ Un ∩ Vn.Then {Wn : n ∈ ω} is a discrete family of open subsets of X, acontradiction. The assertion 2 is proved.

Let U and V be two open subsets of X and F ⊆ V ⊆ clXV ⊆ U .Then {Hn = Un \clXV : n ∈ ω} is a discrete family of open subsetsof X. Thus Hm = ∅ and Un ⊆ U for some m ∈ ω. The assertion 3is proved. �Corollary 2.12. Let X be a normal space. The following asser-tions are equivalent:

1. The space X is densely q-complete.2. The space X contains a dense q-complete subspace.3. The space X is densely fan-complete.4. The space X contains a dense fan-complete subspace.

Corollary 2.13. A space X is densely sieve-complete if and onlyif it contains a dense Cech-complete subspace.

Example 2.14. LetW0 be the set of all countable ordinal numbersand ω1 be the first uncountable ordinal number. In the set V0 =W0×[0, 1) consider the linear order: (α, u) < (β, v) whenever α < βor α = β and u < v. The space V0 with the topology induced by


the linear order is called the long line ([17], Problem 3.12.18). Thespace V = V0∪{ω1}, where x < ω1 for any x ∈ V0, with the topologyinduced by the linear order is the Stone-Cech compactification ofthe space V0. Let Y be the set of all points (α, 0) ∈ V0, where α is anon-limit ordinal number. Obviously, the subspaceX = V0\Y of V0is locally metrizable and locally Cech-complete. Hence X is a sieve-complete space. We affirm that X is not a Gδ-subspace of somepseudocompact space. Really, assume that X is a subspace of apseudocompact space Z and X = Z ∪ (∪{Fn : n ∈ ω}), where {Fn :n ∈ ω} is a sequence of closed subsets of Z. Consider the continuousmappings φ : βX → βZ and ψ : βX → V , where φ(x) = ψ(x) = xfor any x ∈ X. By construction, Φn = φ(ψ−1(ψ(φ−1(clβZFn))))) isa compact subset of βZ\X. We can suppose that Fn = Z∩Φn. Thespace Z1 = βX\ψ−1(ω1) is countably compact and locally compact.If (α, 0) ∈ Y and H(α) = φ(ψ−1(α, 0)), then H(α) ⊆ βZ \X. SinceZ is pseudocompact X∩H(α) = ∅. Thus Z∩H(α) = ∅ andH(α) ⊆Φn(α) for some n(α) ∈ ω. Therefore φ(Z1 \ X) ⊆ ∪{Φn : n ∈ ω}.Consequently, X = ∩{Z1 \ φ−1(Φn) : n ∈ ω} is a Gδ-subset of thespace Z1 and X is Cech-complete, a contradiction (see [2], [17]).

There exists a pseudocompact space X1 such that the space ofrationalsQ is a closed subspace. LetX2 =X×X1. By construction,X2 is a fan-complete space, which is not locally Cech-complete andis not a Gδ-subspace of some pseudocompact space.

3. On quasicontinuous mappings

A mapping f : X×Y → Z of a product space X×Y into a spaceZ is called right strongly quasicontinuous at a point (a, b) ∈ X × Yif for each open neighbourhood W of f(a, b) in Z and every openneighbourhood U of a in X, there exist a non-empty open set U1

in X and an open set V in Y such that U1 ⊂ U , b ∈ V, andf(U1 × V ) ⊂ W . In a similar way one can define a left stronglyquasicontinuous at a point (a, b) ∈ X × Y mapping. A mapping fis strongly quasicontinuous at a point (a, b) ∈ X ×Y if it is left andright strongly quasicontinuous at (a, b) ∈ X × Y .

A mapping f : X → Y of a space X into a space Y is calledquasicontinuous at a point b ∈ X if for each open neighbourhood Vof f(b) in Y and every open neighbourhood U of b in X there existsa non-empty open set W in X such that W ⊂ U and f(W ) ⊂ V .


If a mapping f : X × Y → Z is right or left strongly quasicon-tinuous, then f is quasicontinuous (about quasicontinuity see [7],[19], [20], [22]).

Proposition 3.1. Let X1 be a dense subspace of a space X, Y1 bea dense subspace of a space Y , and f be a separately continuousmapping of X × Y into a space Z, g = f |(X1 × Y1), a ∈ X1, andb ∈ Y1. Then:

1. The mapping f is right (left) strongly quasicontinuous at apoint (a, b) if and only if the mapping g is right (left) strongly qua-sicontinuous at (a, b).

2. The mapping f is quasicontinuous at a point (a, b) if and onlyif the mapping g is quasicontinuous at (a, b).

Proof. Let c = f(a, b) andW be a neighbourhood of c in Z. Fix twoopen subsets W1 and W2 of Z such that c ∈ W2 ⊆ clZW2 ⊆ W1 ⊆clZW1 ⊆ W . Suppose that U is an open subset of X and V is anopen subset of Y such that g(U1 × V1) ⊆ W2, where U1 = U ∩X1

and V1 = V ∩ Y1. For every x ∈ U1, we have f({x} × clY V1) ⊆clZW2 ⊆ W1, by the separate continuity of f . Since V ⊆ clY V1,we have f(U1 × V ) ⊆ clZW2 ⊆∈ W1. For every y ∈ V , we havef(clXU1 × {y}) ⊆ clZW1 ⊆ W , by the separate continuity of f .Since U ⊆ clXU1, we have f(U × V ) ⊆ clZW1 ⊆W . �

The following statement can be found in ([7], Lemma 2.3.7), forthe Cech-complete spaces, and in [23], for strongly β-unfavorablespaces.

Proposition 3.2. Suppose that X is a densely q-complete space,a ∈ X, b is a qD-point of a space Y and f is a separately continuousmapping of X × Y into a space Z. Then:

- f is right strongly quasicontinuous at (a, b);- If X and Y are homogeneous spaces, then f is right strongly

quasicontinuous at every point of X × Y .

Proof. Fix a dense subspace X1 of X and a dense wA-sieve U={γn = {Uα : α ∈ An}, πn : An+1 → An : n ∈ ω} with the Property(C2) on the space X. We can assume that a ∈ X1. Now wefix a dense subspace Y1 of Y and a sequence of neighbourhoods{Hn : n ∈ ω} of the point b in Y such that if yn ∈ Y1 ∩Hn, thenthe sequence {yn : n ∈ ω} has a cluster point in Y . We can assumethat b ∈ Y1 (we take Y1 ∪ {b} in the place Y1, in the contrary). Letg = f |(X1 × Y1).


Put c = f(a, b). Suppose that the mapping f is not right stronglyquasicontinuous at (a, b). By virtue of the Proposition 3.1, themapping g is not right strongly quasicontinuous at (a, b). Thenthere exist a neighbourhood W1 of c and a neighbourhood U of asuch that for each non-empty open in X subset U ′ of U and eachneighbourhood V of b in Y we have f((U ′×V )∩(X1×Y1))\W1 = ∅.Since Z is regular, we can find open neighbourhoods W0 and W ofc such that clZW0 ⊂W ⊂ clZW ⊂W1.

We will construct by induction certain decreasing sequences ofopen sets {Un : n ∈ ω} and {Vn : n ∈ ω} in X and Y, respectively,a c-sequence α = {αn : n ∈ ω} and sequences {xn : n ∈ N} ⊂ X1

and {yn : n ∈ ω} ⊂ Y1 with the following properties:(i) xn ∈ Un ⊆ clXUn ⊆ Uαn ∩ U , yn ∈ Vn ⊆ clY Vn ⊆ Hn,

clXUn+1 ⊆ Un, clY Vn+1 ⊆ Vn, and f(xn, yn) ∈ f(Un × Vn) \W1 ⊆Z \ clZW for each n ∈ ω;

(ii) If M0 = {x ∈ X : f(x, b) ∈W0}, then U0 ⊆M0;(iii) If n ≥ 1, Mn = {x ∈ Un−1 : f(x, yn−1) ∈ Z \ clZW} and

Ln = {y ∈ Vn−1 : f(xn−1, y) ∈ W0}, then clXUn ⊆ Mn, andclY Vn ⊆ Ln.

We proceed as follows.Step 0. Put M0 = {x ∈ X : f(x, b) ∈ W0}. Then M0 is open,

by the separate continuity of f, and a ∈ M0, since f(a, b) ∈ W0.Fix α0 ∈ A0 and an open subset U0 of X such that a ∈ U0 ⊆clXU0 ⊆ M0 ∩ Uα0 ∩ U . Let V0 be an open subset of Y such thatb ∈ V0 ⊆ H0. By the assumption, the set f(U0 × V0) \W1 is notempty. Therefore, we can choose x0 ∈ U0 ∩ X1 and y0 ∈ V0 ∩ Y1such that f(x0, y0) ∈ Z \W1.

Step 1. Put M1 = {x ∈ U0 : f(x, y0) ∈ Z \ clZW}. Thenx0 ∈M1, and M1 is open in X, by the separate continuity of f. Letα1 ∈ π−1

0 (α0) and U1 be an open neighbourhood of x0 in X suchthat clXU1 ⊂ U0 ∩M1 ∩Uα1 . Put L1 = {y ∈ V0 : f(x0, y) ∈W0}. Itfollows from x ∈ U1 ⊂ M0 that b ∈ L1, and that L1 is open in Y,by the separate continuity of f. Let V1 be an open neighbourhoodof b in Y such that clY V1 ⊂ V0 ∩ L1 ∩H1. By the assumption, theset f(U1×V1)\W1 is not empty. Then we can choose x1 ∈ U1∩X1

and y1 ∈ V1 ∩ Y1 such that f(x1, y1) ∈ Z\W1.Step n + 1. Assume that we have already defined the open

sets Un and Vn in X and Y , respectively, an element αn ∈ An andthe points xn ∈ Un and yn ∈ Vn such that Un ⊂M0 ∩ Uαn , b ∈ Vn,


and f(xn, yn) ∈ Z \ clZW . Then we put Mn+1 = {x ∈ Un :f(x, yn) ∈ Z \ clZW}. Clearly, the set Mn+1, is open, and xn ∈Mn+1. Now we let αn+1 ∈ π−1

n (αn), Un+1 be any open neighbour-hood of xn in X such that clXUn+1 ⊆ Mn+1 ∩ Uαn+1 ∩ Un. PutLn+1 = {y ∈ Vn : f(xn, y) ∈W0}. Clearly, this set is open and con-tains the point b, since xn ∈ Un ⊆ M0. Now let Vn+1 be any openneighbourhood of b in Y such that clZVn+1 ⊆ Vn ∩Hn+1 ∩ Ln+1.

Again, by the assumption, the set f(Un+1 × Vn+1) \W1 is notempty, and we can choose points xn+1 ∈ Un+1 ∩ X1 and yn+1 ∈Vn+1∩Y1 such that f(xn+1, yn+1) ∈ Z\W1. Step n+1 is complete.

Let P = ∩{clXUn : n ∈ ω} and H = ∩{clY Vn : n ∈ ω}. Byconstruction, there exist an accumulation point x∗ ∈ P for thesequence { xn : n ∈ ω} in X, and an accumulation point y∗ ∈ Hfor the sequence { yn : n ∈ ω} in Y . Since clXUn+1 ⊂ Un, the pointx∗ belongs to each Un. Now, from x∗ ∈ Un+1 ⊂ Mn+1 it followsthat f(x∗, yn) ∈ Z \ clZW , for each n ∈ ω. Since f is separatelycontinuous, we conclude that f(x∗, y∗) ∈ Z \W .

Fix n ∈ ω. Then yn+1 ∈ Vn+1 ⊂ {y ∈ Y : f(xn, y) ∈ W0}.Therefore, f(xn, ym) ∈ W0 for each m > n. By the separate con-tinuity of f , it follows that f(xn, y

∗) ∈ clZW0. Then again, by theseparate continuity of f , f(x∗, y∗) ∈ clZW0 ⊂ W , a contradiction.The proof is complete. �Corollary 3.3. Suppose that X and Y are densely q-completespaces, and that f is a separately continuous mapping of X × Yinto a space Z. Then there exist a dense subspace X1 of the spaceX and a dense subspace Y1 of the space Y such that the mapping fis strongly quasicontinuous at every point of X1×Y1. In particular,the mapping f is quasicontinuous.

4. On semitopological groups with a quasicontinuousmultiplication

We need the following simple lemma.

Lemma 4.1. Let U be an open neighbourhood of the neutral ele-ment e in a semitopological group G with a right strongly quasicon-tinuous multiplication m : G×G→ G. Then:

- e ∈ clXU∗, where U∗ = {x ∈ G : (x.e) ∈ Int(m−1(U))};

- for each point a ∈ U∗ there exists an open neighbourhood V ofthe neutral element such that aV 2 ⊆ U .


Proof. By the definition, U∗ = ∪{W ⊆ U : U is open in G andW · V ⊆ U for some neighbourhood V of the neutral element e}.Thus the statements follows from right strongly quasicontinuity ofthe multiplication. �

The following lemma was proved in ([23], Lemma 2).

Lemma 4.2. Let G be a semitopological group with a right stronglyquasicontinuous multiplication, X be a dense subspace of G, {H ′

n :n ∈ ω} and {H ′′

n : n ∈ ω} be two sequences of open subsets of G,∩{H ′

n : n ∈ ω} = ∅ and ∩{H ′′n : n ∈ ω} = ∅. Suppose that every

sequence {xn ∈ X ∩ (H ′n ·H ′′

n) : n ∈ ω} has an accumulation pointin G. Then G is a paratopological group.

Proof. We can assume that e ∈ Hn = H ′n = H ′′

n and clGHn+1 ⊆ Hn

for each n ∈ ω.Assume that the multiplication is not jointly continuous. Then

there exists two neighbourhoods W and V of the neutral elementsuch that clGV ⊆ W and U2 \ clGW = ∅ for each neighbourhoodU of the neutral element. We put H = Int(m−1(V )) and V ∗ ={x ∈ G : (x.e) ∈ H}.

We will inductively define two sequences {bn ∈ X : n ∈ ω} and{cn ∈ X : n ∈ ω} of points, and two sequences {Vn : n ∈ ω} and{Wn : n ∈ ω} of open neighbourhoods of the neutral element suchthat:

- ci ∈ V ∗ ∩ Vi−1 ∩X and ci ·Wi ·Wi ⊆ V for each n ∈ ω;- bi ∈W 2

i \ clGW and Vi · bi ⊆ G \ clGW for each n ∈ ω;- Wi+1 ⊆Wi ∩Hi+1 and Vi+1 ⊆ Vi ∩Wi+1 for each n ∈ ω.Step 0. Fix c0 ∈ X ∩ V ∗. Since (c0, e) ∈ H, then there exists a

neighbourhood W0 of e such that W0 ⊆ H0 and c0 ·W 20 ⊆ V . Now

choose a point b0 ∈ X ∩ (W 20 \ clGW ) and a neighbourhood V0 of

e such that V0 ⊆ W0 and V0 · b0 ⊆ G \ clW . Finally fix a pointc1 ∈ V0 ∩X.

Step n+1. Since cn+1 ∈ V ∗ ∩ Vn, then there exists a neighbour-hoodWn+1 of e such thatWn+1 ⊆ Hn+1∩Vn∩Wn and cn+1·W 2

n+1 ⊆V . Now choose a point bn+1 ∈ X∩(W 2

n+1\clGW ) and a neighbour-hood Vn+1 of e such that Vn+1 ⊆Wn+1 and Vn+1 · bn+1 ⊆ G \ clW .Finally, fix a point cn+2 ∈ Vn+1∩X. This completes the induction.

Let b be an accumulation point of the sequence {bn}, and c bean accumulation point of the sequence {cn}. Assume that n < k.


Then ank = cn · bn ∈ cn · Wn · Wk ⊆ cn · W 2n ⊆ V . Therefore,

an = cn · b ∈ clGV , and a = c · b ∈ clGV ⊆ W . Since ck · bn ∈Vk · bn ⊆ G \ clGW , we have a ∈ G \W , a contradiction. �

5. Main results

The next three theorems improve Montgomery Theorem [28] andEllis Theorem [16]; they are closely related to some results in [7],[8], [9], [10], [11], [12], [20], [23], [38].

Theorem 5.1. Suppose that a paratopological group G contains adense fan-complete subspace. Then G is a topological group.

Proof. Assume that G is not a topological group. Then (see [7],Lemma 2.3.19, or [8], Lemma 1.2) there exists an open subset U ofG such that e ∈ U and the set U ∩ U−1 is nowhere dense in G.

Let X be a dense fan-complete subspace of G. There exist asequence {γn = {Vα : α ∈ An} : n ∈ ω} of families of open subsetsof the space G and a sequence {πn : An+1 → An : n ∈ ω} ofmappings such that:

(RC1) X ⊆ ∪{Vα : α ∈ A0} and Vα∩X ⊆ ∪{Vβ : β ∈ π−1n (α)} ⊆

Vα for all α ∈ An and n ∈ ω;(RC2) for any c-sequence α = {αn ∈ An : n ∈ ω}, the sequence

{X ∩ Vαn ;n ∈ ω} is stable in X.Fix β0 ∈ A0 and an open non-empty subset W of G such that

e ∈W and clGW ·W ⊆ U ∩Vβ0 . Put O =W \ clG(U ∩U−1). ThenO ⊆W ⊆ clG(X ∩O) and U ∩O−1 = ∅.

We are going to define a sequence {Un : n ∈ ω} of open subsetsof G, a sequence {xn ∈ X : n ∈ ω} of points, and a c-sequence{αn ∈ An : n ∈ ω} such that:

- xn ∈ X ∩ Un for any n ∈ ω;- xn+1 ∈ Un ∩ xnO for any n ∈ ω;- clGUn+1 ⊆ Un ∩ Vαn+1 ∩O for any n ∈ ω;- α0 = β0 and πn(αn+1) = αn for any n ∈ ω.Let α0 = β0, U0 = O and x0 ∈ X ∩O.Assume that n ∈ ω and αn, Un and xn are already constructed.

Since e ∈ W ⊆ clGO, we have that xn ∈ clGxnO = clG(X ∩ xnO).Thus, Un ∩ xnO ∩ X = ∅. Take xn+1 ∈ X ∩ Un ∩ xnO. Sincexn+1 ∈ X∩Vαn ⊆ ∪{Vβ : β ∈ π−1

n (αn)}, there exist αn+1 ∈ π−1n (αn)

and an open subset Un+1 of G such that xn+1 ∈ Un+1 ⊆ clGUn+1 ⊆Un ∩ Vαn+1 ∩ xnO.


Put F = ∩{clGUn : n ∈ ω}. Obviously, F = ∩{Un : n ∈ ω} =Lim{Un : n ∈ ω} and F = ∅. The set V = F ·W is open in G, andF ⊆ V . Let P = clGV and H = clG(X \ P ) = clG(G \ P ). By theconstruction, P and H are regular closed subsets of G (a regularclosed set is the closure of an open set).

Since F ⊆ V and H ∩ V = ∅, we have: F ∩H = ∅.We claim that H ∩ clGUk = ∅ for some k ∈ ω. Indeed, assume

the contrary. Then Un∩H = ∅ for any n ∈ ω. It follows thatWn =Un ∩ (G \P ) = ∅ and Wn ⊆ Vαn , for any n ∈ ω. Since the sequence{Vαn : n ∈ ω} is stable in G, we have Φ = Lim{Wn;n ∈ ω} = ∅.Hence, Φ ⊆ F ∩ H and F ∩ H = ∅, a contradiction. Therefore,H ∩ clGUk = ∅ for some k ∈ ω.

Thus, Un ⊆ clGUn ⊆ V ⊆ P , for all n ≥ k. Hence xn ∈ V forall k ≥ n. However, F ⊆ Un+2 ⊆ xn+1O ⊆ xn+1W for all k ≥ n.Hence, xk ∈ clGV ⊆ xk+1clGWW . Taking into account that xk+1 ∈xkO, we obtain xk ∈ xkO ·clGWW . Hence, e ∈ O ·clGWW ⊆ O ·U .Therefore, O ∩ U−1 = ∅, a contradiction. �

The next theorem could be derived as a corollary from [23],Theorem 1.

Theorem 5.2. Suppose that a semitopological group G is denselyq-complete. Then G is a topological group.

Proof. Fix a dense subspace X of G and a dense A-sieve U={γn = {Uα : α ∈ An}, πn : An+1 → An : n ∈ ω} with the Property(C2) on the space G. We can assume that e ∈ X and there existsa c-sequence α = {αn : n ∈ ω} such that e ∈ ∩{Uαn : n ∈ ω}.

By virtue of Proposition 3.2, the multiplication m : G×G→ Gis right strongly quasicontinuous at every point of G×G.

We will inductively define a c-sequence β = {βn : n ∈ ω}, asequence of points {bn ∈ X : n ∈ ω}, a sequence {Hn : n ∈ ω} ofopen sets, and a sequence {Vn : n ∈ ω} of open neighbourhoods ofthe neutral element such that:

- b0 ∈ b0 · V0 ⊆ H0 ⊆ Uβ0 and H0 · V0 ⊆ Uα0 ∩ Uβ0 ;- bn · Vn ⊆ Hn for each n ∈ ω;- Vn+1 ⊆ clGVn+1 ⊆ Vn ∩ Uαn+1 for each n ∈ ω;- Hn+1 ⊆ clGHn+1 ⊆ Hn ∩ Uβn+1 for each n ∈ ω;- Hn+1 · Vn+1 ⊆ Uβn+1 ∩Hn for each n ∈ ω.Step 0. Let β0 = α0. Since the multiplication is right strongly

quasicontinuous at (e, e), there exists a neighbourhood W0 of eand an open non-empty subset H0 of Uβ0 such that H0 ·W0 ⊆ Uα0 .


Fix a point b0 ∈ H0 and a neighbourhood V0 of e so that clGV0 ⊆W0

and clG(b0 · V0 ⊆ H0.Step n+1. Fix βn+1 ∈ π−1(βn) such that bn ∈ Uβn+1 . Since

the multiplication is right strongly quasicontinuous at (bn, e), thereexists a neighbourhood Wn+1 of e and an open non-empty subsetHn+1 of Uβn+1 ∩ bnVn such that clG(Hn+1 ·Wn+1) ⊆ Uβn+1 ∩ bnVn.Fix a point bn+1 ∈ Hn+1 and an open neighbourhood Vn+1 of e suchthat clGVn+1 ⊆Wn+1 and clG(bn+1 ·Vn+1) ⊆ Hn+1. This completesthe induction.

By the construction, clG(Hn ·Vn) ⊆ Uβn and any sequence {xn ∈X ∩ (Hn ·Vn) : n ∈ ω} has an accumulation point in G. Lemma 4.2implies that G is a paratopological group. Since every densely q-complete space is densely fan-complete, by Theorem 5.1, it followsthat G is a topological group. �Theorem 5.3. Lef G be a semitopological group of pointwise count-able type. If G is densely q-complete, then G is a Cech-completetopological group.

Proof. It follows from Theorem 5.2 that G is a topological group.Every topological group of pointwise countable type is paracom-pact [7]. Corollary 2.6 implies that G has a dense Cech-completesubspace X.

Denote by ρG the Raikov completion of the topological group G(see [7]). There exists a sequence {Un : n ∈ ω} of open subsets of ρGsuch that X = ∩{Un : n ∈ ω}. By the construction, if Y ⊆ ρG \Gis a dense in ρG subspace, then Y is of the first category. Supposethat c ∈ ρG \ G. Then cX is a dense Cech-complete subspace ofρG, and cX ∩ X = ∅, a contradiction. Thus, G = ρG, that is,G is a Raikov complete group. By Theorem 4.3.15 from [7], G isCech-complete. �Corollary 5.4. For any semitopological group G, the following con-ditions are equivalent:

1. G is a Cech-complete topological group.2. G has a dense Cech-complete subspace.3. G is a Tychonoff space, and there exists a compactification

bG of G such that the remainder bG\G is a set of the first categoryin bG.

4. G is a Tychonoff space, and for any compactification bG of Gthe remainder bG \G is a set of the first category in bG.


5. G is a densely q-complete space of pointwise countable type.6. G is a paratopological group of pointwise countable type, and

G contains a dense fan-complete subspace.7. G is a µ-complete space, and G contains a dense fan-complete

subspace.

Corollary 5.5. Suppose that G is a paratopological group, and thatG is a dense subspace of a feebly compact (or pseudocompact) spaceZ. Suppose further that Z \ G is a set of the first category in Z.Then G is a topological group.

The next statement means that every Cech-complete topologicalgroup is absolutely closed in the class of quasitopological groups.

Corollary 5.6. Let H be a Cech-complete subgroup of a quasitopo-logical group G. Then H is closed in G.

Proof. Let P = clGH. By Proposition 1.4.13 [7], P is a subgroupof G. Theorem 5.3 implies that P is a Cech-complete topologicalgroup. Since H is Raikov complete ([7], Theorem 4.3.15), we haveH = P . The proof is complete. �Theorem 5.7. Lef X be a normal dense subspace of a semitopolog-ical group G. If X is densely fan-complete, then G is a topologicalgroup.

Proof. By virtue of Corollary 2.12, the spaces X and G are dense-ly q-complete. By Theorem 5.2 it follows that G is a topologicalgroup. �

The last statement does not generalize to metrizable paratopo-logical groups.

Example 5.8. By Example 1.4.17 ([7], p. 30), there exists anon-discrete first-countable paratopological group G and a discretecountable subgroup H such that H is not closed in G. Fix a pointb ∈ clGH\H. Denote by B the subgroup of G generated byH∪{b}.Then B is a countable metrizable paratopological group, and H isa non-closed discrete subgroup. Obviously, B is not a topologicalgroup.

Remark 5.9. A. V. Korovin [24] (see also [7], Example 2.4.16) con-structed a commutative quasitopological pseudocompact Booleangroup which is not a topological group. Therefore, a fan-complete


quasitopological group needn’t be a paratopological group. In par-ticular, Theorem 5.1 does not hold for quasitopological groups.

Problem 5.10. Suppose that H is a Raikov complete topologicalgroup, and that H is a subgroup and a subspace of a quasitopologicalgroup G. Is it true that H is closed in G?

By Theorem 5.1 and Proposition 2.9, the answer to Problem2.4.5 in [7] is positive.

6. On analytic spaces with the Baire property

Let J = ωω be the space of irrational numbers, with the usualtopology. If t = (t0, t1, ...) and n ∈ ω, then t|n = t0t1...tn. A family{Et|n : t ∈ J, n ∈ ω} of subsets of a space X is called an array ofsubsets of X.

A subspace Y of a space X is an A-FU -subset of X if there existsan array {Et|n : t ∈ J, n ∈ ω} of X such that Y = ∪{∩{Et|n : n ∈ω} : t ∈ J} and for all t ∈ J and n ∈ ω there exist a closed subsetF ⊆ X and an open subset U of X such that Et|n = F ∩ U .

A space Y is Cech-analytic if Y is an A-FU -subset of some Cechcomplete space (see [18], [10]).

A space Y is a q-analytic if Y is an A-FU -subset of some q-complete space.

A space Y is feebly-analytic if Y is an A-FU -subset of some fan-complete space.

Theorem 6.1. Let Y be a dense A-FU -subset of a space X, andlet Y have the Baire property. Then there exists a dense Gδ-subsetZ of X such that Z ⊆ Y .

Proof. Let Y = ∪{∩{Et|n : n ∈ ω} : t ∈ J} and Et|n = Ft|n ∩ Ut|n,where Ut|n is open in X, Ft|n is closed in X, and Ft|n ⊆ clXUt|n.Then X1 = X \ ∩{clXUt|n \ Ut|n : t ∈ J, n ∈ ω} is a Gδ-subset ofX, and Y1 = Y ∩X1 is dense in X. Thus, we can assume that thesets X1 = X and Et|n are closed in X.

We put Yt0t1...tm = ∪{∩{Et|n : n ∈ ω} : t ∈ J, t|m = t0t1...tm}.We can assume that Ft|n = clXYt|n. We have Yt0t1...tm =∪{Yt0t1...tmn : n ∈ ω}.

Now we will construct open subsets Vt|n of X as follows:


1. Put Vn = IntEn \ ∪{Em : m < n} for each n ∈ ω. The setW0 = ∪{Vn : n ∈ ω} is open and dense in X.2. Assume that n ≥ 0 and that the sets {Vt|n : t ∈ J} have

been already defined. Fix t0, t1, ...tn ∈ ω. We put Vt0t1...tnm =Vt0t1...tn ∩ (IntEt0t1...tnm \ ∪{Et0t1...tnk : k < m}) for each m ∈ ω.

The family {Vt|n : n ∈ ω, t ∈ J} is constructed. The family{Vt|n : t ∈ J} is disjoint, and the set Wn = ∪{Vt|n : t ∈ J} isopen and dense in X, for each n ∈ ω. By the construction, Z =∩{Wn : n ∈ ω} is open and dense in X, and Z ⊆ Y . The proof iscomplete. �

Corollary 6.2. Every Cech-analytic space with the Baire propertycontains a dense Cech complete Gδ-subspace.

Corollary 6.3. Every q-analytic space with the Baire property con-tains a dense q-complete Gδ-subspace.

Corollary 6.4. Every feebly-analytic space with the Baire propertycontains a dense fan-complete Gδ-subspace.

Corollary 6.5. (see [10] for Cech-analytic semitopological groups).Let G be a semitopological group with a dense Cech-analytic sub-space with the Baire property. Then G is a Cech-complete topolog-ical group.

Corollary 6.6. Let X be a dense subspace with the Baire propertyof a semitopological group G. If X is an A-FU -subset of somedensely q-complete space, then G is a topological group.

Corollary 6.7. Let G be a paratopological group. If G contains adense feebly-analytic subspace with the Baire property, then G is atopological group.

Corollary 6.8. Let G be a semitopological group with a densefeebly-analytic subspace Y with the Baire property. Then the fol-lowing conditions are equivalent:

1. The space G is paracompact.2. The space G is µ-complete.3. The space G is Cech-complete.4. G is a space of pointwise countable type.


7. Countable networks in remainders ofsemitopological groups

We apply some results obtained above to study the structureof semitopological groups that have a remainder with a countablenetwork. Here is the main result:

Theorem 7.1. If a non-locally compact semitopological group Ghas a compactification bG such that the remainder X = bG \ Ghas a countable network, then G has a countable π-base and is firstcountable, and the remainder X has a dense separable metrizablesubspace.

To prove this theorem, we need a general lemma on the structureof regular spaces with a countable network.

Lemma 7.2. Suppose that X is a regular space with a countablenetwork S. Then X = Y ∪ Z, where Y is a separable metrizablespace, and Z has a countable network P such that every element ofP is nowhere dense in X.

Proof. Since X is regular, we may assume that each element of S isclosed in X. For eachM ∈ S put PM =M \Int(M), where Int(M)is the interior of M in X. Put P = {PM : M ∈ S}. Clearly, eachelement of the family P is closed and nowhere dense in X.

Let Z be the set of all points ofX at which the countable family Pis a network, and put Y = X \Z. Put also B = {Int(M) :M ∈ S}.

Claim: The family B is a base of X at each point of Y .Indeed, take any y ∈ Y , and let O(y) be an arbitrary open neigh-

bourhood of y in X. Therefore, we can fix an open neighbourhoodV of y in X such that no element of the family P contains y and iscontained in V . Clearly, we may assume that V ⊂ O(y).

Since S is a network of X, there existsM ∈ S such that y ∈M ⊂V . Then PM ∈ P and PM ⊂ V . Now it follows from the choice ofV that y /∈ PM . Since y ∈ M , the definition of PM implies thaty ∈ Int(M). We also have Int(M) ∈ B and Int(M) ⊂ V ⊂ O(y).The Claim is proved.

Hence, Y is a regular space with a countable base, that is, Y isseparable and metrizable [17]. �

We are now in a position to prove Theorem 7.1.


Proof. By Lemma 7.2, X = Y ∪ Z, where Y is a separable metriz-able space, and Z has a countable network P such that every ele-ment of P is nowhere dense in X.

Observe that X is dense in bG as well, since the space G isnowhere locally compact.

Case 1. Y is dense in X. Then Y is dense in bG, since X isdense in bG. Since Y has a countable base, it follows that bG hasa countable π-base. Therefore, G has a countable π-base, since Gis dense in bG.

Case 2. Y is not dense in X. Denote by U the complement in bGof the closure of Y in bG. Then U is a non-empty open subspaceof bG.

For an arbitrary P ∈ P, denote by FP the closure of P in bG.Since P is nowhere dense in X, it follows that FP is nowhere densein bG. Therefore, the set WP = U \ FP is a dense open subspaceof U . Since U is itself open in bG, and bG is compact, it followsthat the subspace H = ∩{WP : P ∈ P} of U is a Cech-completespace dense in U . Now it follows by a standard argument thatG has a dense Cech-complete subspace. Therefore, G is itself aCech-complete topological group. If a topological group G has acompactification such that the remainder has a countable network,then G is a separable metrizable space (see [5]). In this case theremainderX is a Lindelof p-space [4] with aGδ-diagonal and, hence,X is separable and metrizable.

Observe that G is a space of pointwise countable type, since theremainder X is Lindelof [21]. On the other hand, every Tychonoffsemitopological group with a countable π-base has a Gδ-diagonal[8]. Hence, G is first countable [17]. �

Theorem 7.3. If a non-locally compact semitopological group Ghas a separable metrizable remainder, then G is also separable andmetrizable.

Proof. From Theorem 7.1 it follows that G has a countable π-base.Now using results from ([8], Corollary 2.5), we conclude that Ghas a Gδ-diagonal. We also see that G is a Lindelof p-space ([4],Section 3), since G is a remainder of a separable metrizable space.It remains to recall that every Lindelof p-space with a Gδ-diagonalis separable and metrizable (see [17], Problem 5.5.7). �


Example 7.4. Let A be the set of rational numbersQ as a subspaceof Sorgenfrey line S. Then:

- A is a metrizable paratopological group;- A is not a topological group;- A has a metrizable compactification.The paratopological group B from Example 5.8 has the same

properties. The topological spaces A, B, and Q are homeomorphic,but they are not topologically isomorphic as paratopological groups.

Problem 7.5. Is every separable metrizable paratopological (semi-topological) group homeomorphic to a topological group?

Observe in this connection that every countable metrizable semi-topological group is homeomorphic to a topological group (to thegroup of rational numbers in the usual, or discrete topology).

8. On pointwise pseudocompact groups

A point x ∈ X is called a pseudocompactness point of X if thereexists a stable sequence {Un : n ∈ ω} of open subsets of X suchthat x ∈ ∩{Un : n ∈ ω}. A space X is said to be pointwise pseudo-compact if each point of X is a pseudocompactness point (see [7],p. 388). Clearly, every fan-complete space is pointwise pseudocom-pact. Moreover, every q-space is pointwise pseudocompact.

We recall that the Gδ-closure ωclXY of a set Y ⊆ X in a spaceis the set of all points x ∈ X such that every Gδ-set H containingx intersects Y . The set µclXH = ∪{clXA : A ⊆ H,A is a boundedsubset of X} is called the µ-closure of H in X.

The subspace µ∗X = µclβXX of the Stone-Cech compactifica-tion βX of a Tychonoff space X is called the µ-completion of X.Obviously, µ∗X is a subspace of the Dieudonne completion µX ofX.

Let ρG be the Raikov completion of a topological group G andρωG be the Gδ-closure of G in ρG. Clearly, ρωG is a subgroup ofρG.

Lemma 8.1. Let P be a closed bounded subset of a topologicalgroup G. Then ωclρGP = clρGP .

Proof. Let x ∈ clρGP \ ωclρGP . Then there exists a sequence{Un : n ∈ ω} of open subsets of ρG such that x ∈ ∩{Un : n ∈ ω},∩{P ∩ Un : n ∈ ω} = ∅ and clρGUn+1 ⊆ Un for any n ∈ ω.


Then {Vn = G ∩ Un : n ∈ ω} is an open locally finite family ofsubsets of G and P ∩ Vn = ∅ for any n ∈ ω.

A space X is called a paracompact p-space if it admits a perfectmapping onto a metrizable space [1]. A feathered group is a topo-logical group whose underlying space is a paracompact p-space. Atopological group is a feathered group if and only if it is a space ofpointwise countable type (see [7]). �

Theorem 8.2. For any topological group G the following conditionsare equivalent:

1. G is a fan-complete space.2. G contains a dense fan-complete subspace.3. ρωG is a Cech-complete paracompact space.4. G is Gδ-dense in ρG and ρG is a Cech-complete paracompact

space.5. G is a Gδ-subset of the pseudocompact space Z = G ∪ (βG \

µ∗G).

Proof. The implications 1 → 2, 5 → 1 and 3 → 4 → 3 are obvious.Let G be a pointwise pseudocompact topological group. There

exists a stable sequence {Un : n ∈ ω} of open subsets of G and asequence {fn : G → [0, 1] : n ∈ ω} of continuous functions suchthat e ∈ Un+1 = U−1

n+1 ⊆ clGU2n+1 ⊆ Un, Un+1 ⊆ f−1(0) and

G\Un ⊆ f−1(1), for any n ∈ ω. ThenH = ∩{Un : n ∈ ω} is a closedbounded subgroup of G. The closure Φ of H in ρG is a compactsubgroup and the projection φ : ρG → X, where X = ρG/Φ, is aperfect and open continuous mapping onto a complete metrizablespace X.

Fix a complete metric d on a space X. By Theorems 6.9.14 and6.9.15 from [7], we have: ρωG = µG = µ∗G = φ−!(φ(G). FromTheorem 6.5.1 and Corollary 6.9.10 [7] it follows that ρωG is asubgroup of the group ρG.

Assume that G is Gδ-dense in ρG. Then ρωG = ρG = µG = µ∗Gand ρG ⊆ βG. Let Z = G ∪ (βG \ ρG) be a subspace of βG. Bythe construction, G is a Gδ-subset of Z and βZ = βG = βZ.For every Tychonoff space X, the space X ∪ (βX \ µ∗X ⊆ βX ispseudocompact. Hence, Z is a pseudocompact space. Proposition2.3 implies that G is fan-complete. The implications 4 → 1 and4 → 5 are proved.


Assume now that Y is a dense fan-complete subspace of G. ByTheorem 2.5 and Corollary 2.6, there exists a dense Cech-completeparacompact subspace of ρωG. By Theorem 5.3, the topologicalgroup ρωG is Cech-complete. Thus, ρωG = ρG. The implication2 → 4 is proved. The proof is complete. �

Suppose that a topological group G is a q-space. Then in theproof of Theorem 8.2 we can assume that H is a countably compactsubset and {Un : n ∈ ω} is a base of neighbourhoods of the setH in G. In this case the mapping ψ = φ|G : G → Y = φ(G) isquasiperfect, and G is a completeM -space. Moreover, the mappingβψ|Z : Z → βX is quasiperfect, and Z is countably compact. Thus,we have proved the following theorem:

Theorem 8.3. For any topological group G, the following condi-tions are equivalent:

1. G is a complete M -space.2. G is a q-space, and G has a dense fan-complete subspace.3. G is a q-space, and ρωG is a Cech-complete paracompact

space.4. G is a q-space, G is Gδ-dense in ρG, and the space ρG is

Cech-complete and paracompact.5. G is a Gδ-subset of the countably compact space Z = G ∪

(βG \ µ∗G).

Let X be a normal space. If {Un : n ∈ ω} is a stable sequenceof open subsets, then each sequence {xn ∈ Un : n ∈ ω} has a {Un :n ∈ ω} cluster point in X. In particular, each pseudocompactnesspoint is a q-point. Therefore, Theorem 8.2 and Corollary 2.12 yieldthe next result:

Corollary 8.4. Suppose that G is a semitopological group, andthat the space G is normal and has a dense fan-complete subspace.Then:

1. G is a complete M -space.2. G is a topological group.3. G is a Gδ-subset of the countably compact space Z = G ∪

(βG \ µ∗G).

The next example shows that a metrizable topological group withthe Baire property needs not be Cech-complete.


Example 8.5. Let R be the field of reals and Q be the subfieldof rationals. Then there exists an additive subgroup G of R suchthat:

1. G and S = R\G are dense subspaces with the Baire property.2. In G and S every compact subset is countable.3. G ·Q = G.

A subspace M of R is a Q-module if M is an additive subgroupof R and M · Q = M . If L ⊆ R, then we denote by m(L) theQ-module algebraically generated by the set L.

Let {Fα : α < c = 2ω} be the family of all non-countable closedsubsets of R. Assume that F0 = [1, 2].

Using the transfinite recursion, we construct the sequences{xα : α < c} and {yα : α < c} in the following way:

Fix x0 = 2 and y0 ∈ F0 \ Q. Assume that 0 < α < c, and thatLα = {xβ , yβ : β < α} is already defined. Since |m(Lα)| < c, we canfix xα ∈ Fα\m(Lα). LetMα = Lα∪{xα}, and fix yα ∈ Fα\m(Mα).

Now we put X = {xα : α < c}, Y = {yα : α < c}, and Gα =m({xβ : β ≤ α}, for each α < c. Then G = ∪{Gα : α < c} is theQ-module generated by the set X. We claim that G∩Y = ∅. Thus,G is the desired Q-module.

Remark 8.6. There exist a separable complete metrizable linearspace L and a dense linear subspace B of L such that: B andY = L \B are dense subspaces of the space L; B and Y are spaceswith the Baire property; B and Y are not complete metrizablespaces.

Really, let ξ ∈ βω \ ω and X = {ξ} ∪ ω. We put L = RX andB = Cp(X) ⊆ L. D.J. Lutzer and R.A. McCoy [26] have provedthat the space B is not complete metrizable and has the Baireproperty. By virtue of the recent result from ([6], Theorem 1.1),since the space B is not Cech-complete, the subspace Y has theBaire property.

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Department of Mathematics, Ohio University, Athens, OH 45701,USA

E-mail address: [email protected]

Department of Mathematics, Tiraspol State University, Chisinau,Republic of Moldova, MD-2069

E-mail address: [email protected]

topology proceedings 37 (2011) pp. 33-60: completeness...

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