A note on paratopological groups

A note on paratopological groups

Chuan Liu

Abstract. In this paper, it is proved that a first-countable paratopological group has a regularGδ-diagonal, which gives an affirmative answer to Arhangel’skii and Burke’s question [Spaces with a regularGδ-diagonal, Topology Appl. 153(2006), 1917–1929].

IfGis a symmetrizable paratopological group, thenGis a developable space. We also discuss copies ofSω and of S₂ in paratopological groups and generalize some Nyikos [Metrizability and the Fr´echet-Urysohn property in topological groups, Proc. Amer.

Math. Soc. 83(1981), no. 4, 793–801] and Svetlichnyi [Intersection of topologies and metrizability in topological groups, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4(1989), 79–81] results.

Keywords: paratopological group, symmetrizable spaces, regularGδ-diagonal, weak ba- ses, Arens space

Classification: Primary 54H13, 54H99

1. Introduction

Recently, paratopological groups have been studied by many topologists ([3], [4], [19]). It is natural to ask what results on topological groups are valid on paratopological groups. In this paper, by discussing copies of Sω and of S₂ on paratopological groups, we generalize some results from [14], [15] and [18]. We also discuss first-countable paratopological groups and prove that a first-countable paratopological group has a regularG_δ-diagonal, and give an affirmative answer to a question from [3].

Recall that a paratopological group is a group with a topology such that the multiplication is jointly continuous.

All spaces are regularT₁ unless stated otherwise. Ndenotes natural numbers and edenotes the neutral element of a group. We refer to [6] for notations and terminology not given explicitly.

2. Main results

A spaceX is said to have aregular G_δ-diagonal if the diagonal ∆ ={(x, x) : x∈X}can be represented as the intersection of the closures of a countable family of open neighborhoods of ∆ inX×X. According to Zenor [21], a spaceX has a regularG_δ-diagonal if and only if there exists a sequence{G_n:n∈ω} of open covers ofX with the following property:

(*) For any two distinct pointsyandzinX, there are open neighborhoodsOy

andOz ofy andz, respectively, andk∈ω such that no element ofGn intersects bothOx and Oy.

In [3], Arhangel’skii and Burke proved that every Hausdorff first countable Abelian paratopological group G has a regular G_δ-diagonal. We sharpen the result by showing the following

Theorem 2.1. LetGbe a Hausdorff first-countable paratopological group. Then Ghas a regularG_δ-diagonal.

Proof: Fix a countable base{Vn :n ∈N} at the neutral element ein Gwith V_n+1² ⊂Vn. Letx∈G; thenxVn, Vnxare open forn∈NsinceGis a paratopo- logical group. For x ∈ G, n ∈ N, let Wn(x) = xVn∩Vnx. Then Wn(x) is a neighborhood ofx. LetGn={Wn(x) :x∈G}forn∈N. Then{Gn:n∈N} is a sequence of open coverings ofG.

By Zenor’s characterization of regularG_δ-diagonal, we only prove the following Claim: Fory, z∈G,y6=z, there isk∈Nsuch that no element of G_k intersects bothyV_kandzV_k.

Suppose not; for any n ∈ N, there is an element Wn(xn) ∈ Gn such that yVn∩Wn(xn)6=∅andWn(xn)∩zVn6=∅. Then there arean, bn, cn, dnandfnin Vn such that yan=xnbn,xncn=dnxn=zfn, yan =d⁻¹_n dnxnbn=d⁻¹_n zfnbn. Sincean→e, we haveyan→y, henced⁻¹_n zfnbn→y. dn→esincedn∈Vn,G is a paratopological group, thendnd⁻¹_n zfnbn→ey=y, hencezfnbn→y. Notice that fn, bn∈Vn, thus fnbn→e, hencezdnbn→z. Gis Hausdorff, then y =z, this is a contradiction.

Therefore,Ghas a regularG_δ-diagonal.

A subsetAof a spaceX is said to bebounded [3] inX if every infinite family ξof open subsets ofX such thatV ∩A6=∅, for everyV ∈ξ, has an accumulation pointX. IfX is bounded in itself, then we say thatX ispseudocompact.

Notice that a pseudocompact or bounded subset of a regular spaceX is metriz- able ifX has a regularGδ-diagonal [3]. We have the following

Corollary 2.1. LetGbe a regular first-countable paratopological group. Then every pseudocompact subspace of Gis a metrizable compactum.

Corollary 2.2. LetGbe a regular first-countable paratopological group. Then every bounded subspace of Gis metrizable.

The above theorem and corollaries give an affirmative answer to Arhangel’skii and Burke’s question [3, Problem 25].

A spaceX is anw∆-space [8] if there exists a sequence (Gn) of open covers of X such that if xn ∈st(x,G_n) for each n ∈ N, then the set {x_n : n∈ N} has a cluster point inX.

Since a space X with a regular G_δ-diagonal has aG^∗_δ-diagonal, by [8, Theo- rem 3.3], we have the following

Corollary 2.3. Let Gbe a first-countable paratopological group. Then Gis a Moore space if Gis anw∆-space.

A spaceX isquasi-developable [8] if there exists a sequence (G_n) of families of subsets ofX such that for each x∈X, {st(x,G_n) :n∈N} is a base atx. Recall that a topological space is said to besymmetrizable if its topology is generated by asymmetric, that is, by a distance function satisfying all the usual restrictions on a metric, except for the triangle inequality [1].

Theorem 2.2. Every symmetrizable paratopological groupGis a Moore space.

Proof: We fix a symmetric d on the paratopologcial group G generating the topology onG. Since Gis weakly first-countable [1], by a result of Nyikos [15], G is first-countable. Put B(x,1/n) = {y ∈ G : d(x, y) < 1/n}, and fix an open base {Vn : n ∈ N} at e with Vn ⊂ int(B(e,1/n)) and V_n+1² ⊂ Vn. Let Aij ={x∈G:Vix⊂int(B(x,1/j))}andGij ={Vix:x∈Aij}fori, j∈N. Since {Vix:i∈N} and{int(B(x,1/j)) :j∈N}are bases atx, G=S

{Aij :i, j∈N}.

We prove that {st(x,Gij) : i, j ∈ N} is a base at x ∈ G. Let U be an open subset ofX withx∈ U. There exists k∈ Nsuch that x∈int(B(x,1/k))⊂U and pickm, n ∈N such that m < n, Vnx⊂ Vmx ⊂int(B(x,1/k)). We choose k^′ such that B(x,1/k^′) ⊂ Vnx since {B(x,1/i) : i ∈ N} is a weak base at x.

For x ∈ Vny ∈ G_nk′, since Vny ⊂ B(y,1/k^′), d(x, y) = d(y, x) < 1/k^′, hence y ∈ B(x,1/k^′) ⊂ Vnx. Vny ⊂ VnVnx ⊂ Vmx ⊂ int(B(x,1/k)) ⊂ U, hence x∈st(x,G_nk′)⊂U. ThereforeGis quasi-developable.

Gis symmetrizable and first-countable, henceG is semi-stratifiable [8, Theo- rem 9.8], thus every closed subset ofGis aG_δ-set. ThereforeGis a developable

space [8, Theorem 8.6].

We cannot replace “symmetrizable” with “first-countable” in Theorem 2.2, Sorgenfrey line is a first-countable paratopological group but not a Moore space.

Let Sκ be the quotient space obtained by identifying all limit points of the topological sum ofκmany convergent sequences. Sωis called sequential fan. The Arens’ spaceS₂={∞} ∪ {xn:n∈N} ∪ {xn(m) :m, n∈N}is defined as follows:

Each xn(m) is isolated; a basic neighborhood of xn is{xn} ∪ {xn(m) :m > k, for some k∈ N}; a basic neighborhood of ∞ is {∞} ∪(S

{Vn : n > k for some k∈N}), whereVn is a neighborhood ofxn.

In [14], it was proved that a topological group contains a (closed) copy ofSωif and only if it contains a (closed) copy ofS₂. We do not know if the result is still true for paratopological groups, but we have the following theorem by modifying Lemma 2.1 in [14].

Theorem 2.3. Let Gbe a paratopological group. Then G contains a (closed) copy of Sω if Ghas a(closed)copy of S₂.

Proof: Let A={e} ∪ {xn : n∈N} ∪ {xn(m) : m, n∈N} be a closed copy of S₂, where eis the neutral element of G. Forn, m∈N, letyn(m) =x⁻¹_n xn(m).

Then yn(m) → e as m → ∞ for n ∈ N. For each n, let Sn = {yn(m) : m ∈ N}. Then F = {n : Sm ∩Sn is infinite } is finite (otherwise, pick distinct x⁻¹_nixni(m_i)∈Sm∩Sni forn_i ∈ F with n_i < n_i+1, x⁻¹_nixni(m_i)→e, xni →e, hence xni(m_i) → e, a contradiction). Without loss of generality, we assume S_i∩S_j =∅ ifi6=j. LetB={e} ∪ {yn(m) :n, m∈N}.

Claim: B is a closed copy of Sω.

SupposeB is not closed. Then there isx∈X\Bwithx∈B. SinceAis closed, there exists an open neighborhoodV of the neutral elementesuch thatV xmeets {x_n(m) : m ∈ N} for at most one n. Let U be open neighborhood of e with U² ⊂V; U x contains an infinite subset {yni(mi) : i ∈N} ofB. Since xn →e, without loss of generality,{xni:i∈N} ⊂U. {xniyni(mi) :i∈N} ⊂U U x⊂V x, it means{xni(mi) :i∈N} ⊂V x, a contradiction.

Iff :ω →ω, then C=S

{yn(m) :m≤f(n), n∈N} does not have a cluster point. Otherwise, there existsx∈C\{x}. Let V be an open neighborhoodV of the neutral elementesuch thatV xmeets|V x∩ {x_n(m) :m≤f(n), n∈N}| ≤1.

Let U be open neighborhood of e with U² ⊂V, U x contains an infinite subset {yni(mi) :i∈N} ⊂C, hencexni(mi) =xniyni(mi)∈U U x⊂V xfor eachi∈N, which is a contradiction. HenceB is a copy of Sω. Nogura, Shakhmatov and Tanaka proved the following corollary as G is a topological group [14]. By Theorem 2.3, we can see the following corollary is still true for a paratopological groupG.

Note that a sequential space is anA-space¹ if and only if it contains no closed copy ofSω [20]. By Theorem 2.3, a paratopological group contains no closed copy of S₂ if it is an A-space. A sequential space that each point is a G_δ-set or is hereditarily normal is strongly Fr´echet if it contains no closed copy ofSω andS₂ [20, Theorem 3.1]. A strongly Fr´echet space is anα₄-space² [2, Theorem 5.26].

Corollary 2.4. Suppose thatGis a sequential paratopological group such that either(a) e∈Gis aG_δ-set, or(b)Gis hereditarily normal. Then the following

1A spaceX is anA-space if, whenever{An :n∈ N} is a decreasing sequence of subsets ofX, andx∈X is a point withx∈^T{An\{x}:n∈N}, then for everyn∈None can find a (possibly empty) setBn⊂Ansuch that^S{Bn:n∈N}is not closed inX.

2A countable collection{Sn:n∈N}of convergent sequences in a spaceX is called asheaf (with a vertexx) if each sequence Sn converges to the same pointx ∈X. A space is called α₄-space, if for every pointx∈X and each sheaf{Sn:n∈N}with the vertexx, there exists a sequence converging toxwhich meets infinitely many sequencesSn.

are equivalent:

(1) Gis anα₄-space;

(2) Gis anA-space, and (3) Gis strongly Fr´echet.

A paratopological groupGis said to have the property (**), if there exists a sequence {xn : n∈ N} ⊂G such that xn → e and x⁻¹_n → e. Obviously, every topological group has the property (**). Not every paratopological group has the property (**), for instance, Sorgenfrey lineSdoes not have the property (**).

A paratopological group having the property (**) need not be a topological group:

for instance, if (R,+) is the real line with the usual topology, then S×R is a paratopological group having the property (**) but not a topological group.

Theorem 2.4. LetGbe a paratopological group having the property(**). Then Ghas a(closed)copy of S₂ if it has a(closed)copy of Sω.

Proof: Let A = {e} ∪ {yn(m) : m, n ∈ N} be a closed copy of Sω, for each n, yn(m) → e as m → ∞. Since G has the property (**), there is a sequence {xn:n∈N} such that xn →e andx⁻¹_n →e. LetUn be an open neighborhood ofxn for eachnwithUi∩Uj =∅ ifi6=j. Letxn(m) =xnyn(m) for n, m∈N. For anyn∈N, we havexn(m)→xn as m→ ∞. Without loss of generality, we assume{xn(m) :m∈N} ⊂Un. LetB={e}∪{xn:n∈N}∪{xn(m) :n, m∈N}.

Claim: B is a closed copy of S2.

Suppose B is not closed. Then there exists x /∈ B, e 6= x ∈ B\{x}. Since A is closed, there is a neighborhood of e such that V x∩(A\{x}) = ∅. Let U be a neighborhood of e with U² ⊂ V and U x contains at most one xn. U x contains infinitely many elements of B, since U contains infinitely manyx⁻¹_n ’s, U U xcontains infinitely manyyn(m). HenceV xcontains infinitely many elements ofA, this is a contradiction.

Iff :ω→ω, similarly as in the proof of Theorem 2.3,{x_n(m) :n≥kfor some k, m≤f(n)} is closed. HenceB is a closed copy ofS2. Note that a Fr´echet-Urysohn space contains no closed copy of S₂, then a Fr´echet-Urysohn paratopological group having the property (**) contains no closed copy of Sω by Theorem 2.4, hence it is a strongly Fr´echet space [20] (or countably bisequential space [13]), therefore it is anα₄-space [2, Theorem 5.23].

Corollary 2.5. LetGbe a paratopological group with the property(**). If G is a Fr´echet-Urysohn space, thenGis a α₄-space.

Corollary 2.5 gives a partial answer to Nyikos’ question [15, Problem 3]: “Is a Fr´echet-Urysohn paratopological group anα₄-space?”.

Question2.1. Can we omit the property (**) in Theorem 2.4 or in Corollary 2.5?

A space X is called weakly quasi-first countable or ℵ₀-weakly first-countable ([17], [18]) if for eachi∈N, there exists a mappingBⁱ :N×X → P(X), where P(X) denotes the power set ofX, such that the following (1) and (2) hold:

(1) for i ∈ N, for each n ∈ N and x ∈ X, Bⁱ(n+ 1, x) ⊂ Bⁱ(n, x), and {x}=T

{Bⁱ(n, x) :n∈N}; and

(2) a subsetV of X is open if and only if for eachy∈V and for each i∈N there existsn(i) withBⁱ(n(i), y)⊂V.

IfBⁱ =B fori∈N, thenX is calledweakly first countable or g-first countable.

Obviously, a weakly first countable space is weakly quasi-first countable.

Corollary 2.6. LetGbe a Fr´echet-Urysohn paratopological group with the pro- perty(**). If Gisℵ₀-weakly first-countable, thenGis first-countable.

Proof: By Corollary 2.5, G is an α₄-space, hence G is weakly first-countable

[10], thusGis first-countable [15, Theorem 2].

By Corollary 2.5, we have the following:

Corollary 2.7 ([18]). A Fr´echet-Urysohn,ℵ₀-weakly first-countable topological group is metrizable.

Next, we discuss when we cannot embed a copy ofSω₁ to some paratopological group.

A family{Bα:α∈I}of subsets of a spaceX ishereditarily closure-preserving (weakly hereditarily closure-preserving [5]) (simply, HCP (wHCP)) if

[{Cα:α∈J}= ([

{Cα:α∈J})({xα:α∈J} is closed discrete), wheneverJ ⊂IandCα⊂Bα(xα∈Bα) for eachα∈J. Obviously, a HCP family is wHCP. Spaces with a σ-wHCP weak base (base) were discussed in [11], [12].

LetP be a cover of a spaceX. ThenP is ak-network forX if wheneverK⊂U withK compact andU open inX,K ⊂SP^′ ⊂U for some finiteP^′ ⊂ P. A k- network is a network. A space with aσ-locally finite k-network is anℵ-space [16].

Sω₁ is a closed image of a metric space, hence it has a σ-HCP closed k-network [7] but it is not anℵ-space [9].

Theorem 2.5. Let G be a paratopological topological group with the proper- ty (**). If G has a σ-wHCP closed k-network, then Gcontains no closed copy of Sω₁.

Proof: SupposeGcontains a closed copy ofSω₁ ={e}∪{xn(α) :α < ω₁, n∈N}, wheree is the neutral element ofGand xn(α)→e as n→ ∞. SinceGhas the property (**), there exists a sequence {xn : n ∈ N} ⊂ G such that xn → e, x⁻¹_n → e. G is regular, we take open subsets Un of G such that xn ∈ Un, Un∩Um = ∅ (n 6= m) and Un∩ {xn : n ∈ N} = {xn}. For each m ∈ N, xmxn(α)→ xm(n → ∞), {x_mxn(α) : n ∈ N} is eventually in Um for α < ω₁. Without loss of generality, we assume{x_mxn(α) :n∈N} ⊂Um.

Claim: B = {x_n(α)x_m(α)(α) : α < ω₁} is a discrete subset of G for n(α), m(α)∈N.

Case 1: {n(α) :α < ω₁}is finite.

We rewrite {n(α) : α < ω₁} = {r₁, . . . r_k}. Since {x_g(α)(α) : α < ω₁} is discrete for every g : ω1 → N, then {x_rix_g(α)(α) :α < ω1} is discrete for each i≤k, henceB is discrete.

Case 2: {n(α) :α < ω₁}is infinite.

Suppose B is not discrete and let x be the cluster point of B. For every g:ω₁ →N, there exists an open neighborhoodV ofesuch that|V x∩ {x_g(α)(α) : α < ω₁}| ≤ 1. Let U be an open neighborhood of e with U² ⊂ V. Then C=U x∩ {x_n(α)x_m(α)(α) :α < ω1} 6=∅ for infinitely manyn(α). Since x⁻¹_n → e, {xn : n ∈ N} is eventually in U, {x⁻¹_n : n ≥ k}C ⊂ U U x ⊂ V x. Then

|V x∩ {x_g(α)(α) :α < ω₁}| ≥ω, a contradiction.

For α < ω1, let Cα = {e} ∪ {xn : n ∈ N} ∪ {xnxi(α) : n ∈ N, i ≥ fn(α)}.

Note that xnx_jn(α) → e(n → ∞), where jm ≥ fm(α). Since every infinite subset of Cα has a cluster point in it, Cα is a countably compact. Since every countably compact space with a σ-wHCP network has a countable network [12, Proposition 6],Cα is compact [11].

LetP =S

{Pn:n∈N} be aσ-wHCP k-network consisting of closed subsets.

Then there is a finite P^′ ⊂ P such that C₀ ⊂ S

P^′. Pick P₀ ∈ P^′ so that P₀ containsk₀ =x_n(0)x_m(0)(0) and infinitely manyxn’s. We assume that for each α < β, there exists Pα ∈ P such that Pα contains infinitely many xn’s and a pointkα =x_n(α)x_m(α)(α). We haveC_β ⊂G\{kα :α < β}, which is open inG by the Claim. There is a finiteP^′′⊂ P such thatC_β ⊂S

P^′′⊂G\{kα:α < β}, pickP_β ∈ P^′′ so that P_β contains infinitely many xn and k_β =x_n(β)x_m(β)(β).

By induction, we obtain{P_α:α < ω₁} ⊂ P such thatPα6=P_β ifα6=β and each Pα contains infinitely manyxn’s, hence there are uncountably manyPα ∈ P_nfor somen∈N. Note thatP_nis wHCP and there is a subsequenceLof{x_n:n∈N}

such thatLis discrete, which is a contradiction.

References

[1] Arhangel’skiˇı A.V.,Mappings and spaces, Russian Math. Surveys21(1966), 115–162.

[2] Arhangel’skii A.V.,The frequency spectrum of a topological space and the product opera- tion, Trans. Moscow Math. Soc.2(1981), 163–200.

[3] Arhangel’skii A.V., Burke D., Spaces with a regular Gδ-diagonal, Topology Appl. 153 (2006), 1917–1929.

[4] Arhangel’skii A.V., Reznichenko E.A.,Paratopological and semitopological groups versus topological groups, Topology Appl.151(2005), 107–119.

[5] Burke D., Engelking R., Lutzer D.,Hereditarily closure-preserving collections and metriza- tion, Proc. Amer. Math. Soc.51(1975), 483–488.

[6] Engelking R.,General Topology, PWN, Warszawa, 1977.

[7] Foged L.,A characterization of closed images of metric spaces, Proc. Amer. Math. Soc.

95(1985), 487–490.

[8] Gruenhage G.,Generalized metric spaces, in: K. Kunen, J.E. Vaughan eds., Handbook of Set-theoretic Topology, North-Holland, 1984, pp. 423–501.

[9] Gruenhage G., Michael E., Tanaka Y.,Spaces determined by point-countable covers, Pacific J. Math.113(1984), 303–332.

[10] Liu C.,On weakly bisequential spaces, Comment Math. Univ. Carolin.41(2000), no. 3, 611–617.

[11] Liu C.,Notes on g-metrizable spaces, Topology Proc.29(2005), no. 1, 207–215.

[12] Liu C.,Nagata-Smirnov revisited: spaces withσ-wHCP bases, Topology Proc.29(2005), no. 2, 559–565.

[13] Michael E.,A quintuple quotient quest, General Topology Appl.2(1972), 91–138.

[14] Nogura T., Shakhmatov D., Tanaka Y.,α₄-property versusA-property in topological spaces and groups, Studia Sci. Math. Hungar.33(1997), 351–362.

[15] Nyikos P., Metrizability and the Fr´echet-Urysohn property in topological groups, Proc.

Amer. Math. Soc.83(1981), no. 4, 793–801.

[16] O’Meara P.,On paracompactness in function spaces with the compact open topology, Proc.

Amer. Math. Soc.29(1971), 183–189.

[17] Sirois-Dumais R.,Quasi- and weakly-quasi-first-countable space, Topology Appl.11(1980), 223–230.

[18] Svetlichnyi S.A.,Intersection of topologies and metrizability in topological groups, Vestnik Moskov. Univ. Ser I Mat. Mekh.4(1989), 79–81.

[19] Reznichenko E.A.,Extensions of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topology Appl.59(1994), 233–244.

[20] Tanaka Y.,Metrizability of certain quotient spaces, Fund. Math.119(1983), 157–168.

[21] Zenor P.,On spaces with regularGδ-diagonals, Pacific J. Math.40(1972), 759–763.

Department of Mathematics, Ohio University-Zanesville Campus, Zanesville, OH 43701, USA

(Received November 28, 2005)