Two Notes on Topological Groups

ZHANG Ke-xiu，LIU Xin，TANG Zhong-bao

（School of Mathematics and Statistics，Minnan Normal University，Zhangzhou 363000，China）



Two Notes on Topological Groups

ZHANG Ke-xiu，LIU Xin，TANG Zhong-bao

（School of Mathematics and Statistics，Minnan Normal University，Zhangzhou 363000，China）

In this short paper，we firstly give a short proof of Birkhoff-Kakutani Theorem by Moore metrizable Theorem.Then we prove that G is a topological group if it is a paratopological group which is a dense Gδ-set in a locally feebly compact regular space X.

topological group；first-countable；paratopological group；locally feebly compact；metrizable

2000 MR Subject Classification:22A30，54D10，54E99，54H99

Article ID:1002—0462（2016）01—0082—05

Chin.Quart.J.of Math.

2016，31（1）：82—86

§1.　Introduction

No separation restrictions on the topology of the groups are imposed unless we mention them explicitly.We say a space is regular if it is T1and T3.The symbol e denotes the neutral element of a group.We denote by ω the first infinite order.We always writefor the closure of a subspace A in a space X.The readers may consult［3，5］for notation and terminology not explicitly given here.

Recall that a paratopological group G is a group G with a topology such that the product map of G×G into G is continuous.A topological group G is a group G with a（Hausdorff）topology such that G is a paratopological group and the inverse map of G onto itself associating x-1with arbitrary x∈G is continuous.Paratopological groups were discussed and many results have been obtained in［2-3，9-14］.

A topological space G is said to be a feebly compact space if every locally finite family of open sets in G is finite.A topological space G is said to be a locally feebly compact space，if for each x∈G，there exists an open neighborhood V of x such that the closure of V is feebly compact.

The following two theorems are well-known in topological algebra.

Theorem 1.1［4，7］（Birkhoff-Kakutani Theorem）Each T0first-countable topological group is metrizable.

Theorem 1.2［2］Let G be a paratopological group.If G is a dense Gδ-set in a feebly compact regular space X，then G is a topological group.

It is well-known that Birkhoff-Kakutani proved Theorem 1.1 by using prenorm.In this paper，we shall give a short proof of Theorem 1.1 by Moore metrizable Theorem.Moreover，we shall prove that G is a topological group if it is a paratopological group which is a dense Gδ-set in a locally feebly compact space X，which improves Theorem 1.2.

§2.Main Resuts

Let U=｛Us｝s∈Sbe a family of covers of a set G and let M be a subset of G.We always denote st（M，U）by the set∪｛Us：M∩Us≠∅｝.

Theorem 2.1［5］（Moore Metrizable Theorem）

A T0-space X is metrizable if and only if there exists a sequence｛U：∈ω｝of open covers of X such that for each open set U and x∈U，there exists an open set V and n∈ω such that x∈V and st（V，U）⊂U.

Next we shall give a short proof of Theorem 1.1 by Theorem 1.2.

Proof of Theorem 1.1Since G is a first-countable topological group，we can find a local base｛Vn｝n∈ωof the neutral element e of G satisfying the following properties

For each n∈ω，put U=｛V：∈G，∈ω｝.Obviously，｛U｝∈ωis a sequence of open covers of G.

By Theorem 2.1，it suffices to show that for each open set U and x∈U there exists an open set V and n∈ω，such that x∈V and St（V，U）⊂U.

Indeed，since｛xVn：n∈ω｝is a local base at x，there exists n∈ω，such that x∈xVn⊂U. Let V=xVn+1.We claim that st（V，U+）⊂U.Indeed，if z∈St（V，U+），then there exists apoint y∈G such that z∈yVn+2and yVn+2∩V≠∅.Thenand thus

Finally，we recall some lemmas in order to improve Theorem 1.2.

Lemma 2.2［2］Suppose that G is a para topological group and not a topological group. Then there exists an open neighborhood U of the neutral element e of G such that U∩U-1is nowhere dense in G，that is，the interior of the closure of U∩U-1is empty.

A subset M of a space is called a regular closed subset if there exists an open subset U of G such that

Lemma 2.3［15］The following are equivalent for a space X

（1）X is feebly compact；

（2）Ifis a countable collection of open sets with the finite intersection property，then

Lemma 2.4Let G be a feebly compact space.If H is a regular closed subset of G，then H is also feebly compact.

Proof Let C be a countable collection of open sets with the finite intersection property in H.Put C=｛U〉：〉∈ω｝，where each Uiis an open set in H.Then there exists in G a family of open sets｛Ui´：i∈ω｝such that Ui´∩H=Uifor each i∈ω.Obviously，the family｛Ui´：i∈ω｝has the finite intersection property in G.

Since H is a regular closed subset of G，we can find an open set V in G such thatObviously，we haveTherefore，it is easy to see that｛Ui´∩V：i∈ω｝has the finite intersection propertyby Lemma 2.3.Obviously，for all i∈ω.Moreover，.ThusHence H is feebly compact by Lemma 2.3.

Theorem 2.5Suppose that G is a para topological group such that G is a dense Gδ-set in a locally feebly compact regular space X.Then G is a topological group.

Firstly，we fix a sequence｛Mn：n∈ω｝of open sets in X such that G=∩Mn，and for each x∈X，we can find an open neighborhood Vxof x in X such thatis feebly compact. Next we are going to define a sequence｛Un：n∈ω｝of open subsets of X and a sequence｛xn：n∈ω｝of elements of G such that xn∈Unfor each n∈ω andwhenever j＜i. Since O is open in G and G⊂X，there exists an open subsetPick an arbitrary pointand then Put

Suppose that，for some n∈ω，an open subset Unin X and a point xn∈Un∩G are already defined.Sinceit is easy to see thatMoreover， we have Un∩xnO≠∅since Unis an open neighborhood of xnin X.Pick an arbitrary point xn+1∈Un∩xnO.Note that xn+1∈G，since xnO⊂G.Moreover，xn+1∈Un∩Mn∩Vxn+1.

Since X is regular，we can find an open neighborhood Un+1of xn+1in X such that Un+1∩Therefore，the definition of the sets Unand points xn， for each n∈ω，is complete.It is easy to see thatwhenever j＜i and xn+1∈xnO for each n∈ω.

By Lemma 2.4，we claim that H∩F=∅.

Indeed，assume the contrary and fix x∈F∩H.Since FW is an open neighborhood of F in G，it follows from x∈F that there exists an open neighborhood V of x in X such that V∩G⊂FW.Then the density of G in X implies that V⊂P.However，x∈V∩H implies that VP≠∅，which is a contradiction.Therefore，we have H∩F=∅.

Since H is feebly compact，it follows from H∩F=∅and the definition of F that Uk∩H=∅ for someHence we have Uk⊂P，thensince xk∈Uk∩G.However，we have F⊂Uk+2∩G⊂xk+1O⊂xk+1W.Therefore，we have

It follows from xk+1∈xkO that xk∈xkOU.Therefore，we have e∈OU and O-1∩U 6=∅，which is again a contradiction.

Therefore，G is a topological group.

Corollary 2.6If G is a locally pseudo compact para topological group then it is a topological group.

Remark By A V Arhangelskiˇı and M M Choban’s results in［1］，we also can obtain Theorem 2.5.However，our method is different from［1］.

Acknowledgements The authors wish to thank two reviewers for careful reading preliminary version of this paper and providing many valuable suggestions.

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O189.1Document code:A

date：2014-10-09

Supported by the National Natural Science Foundation of China（11201414，11471153）

Biographies：ZHANG Ke-xiu（1978-），female，native of Linyi，Shandong，a lecturer of Minnan Normal University，engages in general topology；LIU Xin（1986-），male，native of Xinyang，Henan，a postgraduate of Minnan Normal University，engages in general topology；TANG Zhong-bao（1984-），male，native of Hengyang，Hunan，a postgraduate of Minnan Normal University，engages in general topology.