# Notes on remainders of paratopological groups.

1 IntroductionBy a remainder of a Tychonoff topological space G, we mean the subspace bG \ G of some compactification bG of G. Remainders of a topological group or a paratopological group have many interesting properties and have been studied extensively in literature (see [1]-[6] and [8]-[11]).

One of the most interesting questions in the study of remainders is to determine to what extent a property of a topological space X is related to another property of some or all remainders of X. A classical result about remainders is the following theorem due to M. Henriksen and J. Isbell [18]:

Theorem 1.1. A Tychonoff space X is of countable type if and only if the remainder in any (or some) Hausdorff compactification of X is Lindelof.

In [11] Arhangel'skii studied properties of topological groups with an Ohio complete remainder. He proved that a non-locally compact topological group G has a remainder which is a p-space if and only if G is either a Lindelof p-space or a [sigma]-compact space.

In this paper we generalize the above result and prove the following: A non-locally compact paratopological group G has a remainder which is a p-space if and only if G is either a Lindelof p-space or a ([sigma]-compact space. We investigate local metrizability of remainders of a paratopological group. We prove that if G is a non-locally compact paratopological group with a compactification bG such that the remainder bG \ G is locally metrizable, then both G and bG are separable and metrizable.

We show that a cosmic paratopological group with a paracompact remainder must be separable and metrizable.

We also investigate remainders of semitopological groups. It is proved that if a separable semitopological group G has a remainder Y with countable [pi]-character, then either Y is countably compact, or G has a countable [pi]-base.

Throughout this paper, a topological space always means a Tychonoff space. c(X) is the cellularity or Souslin number of the space X. For unexplained terms and symbols we refer the reader to [7] or [14].

2 Preliminaries

Recall that a topological group G is a group G with a topology such that multiplication on G considered as a map of G x G to G is jointly continuous and the inversion in G is continuous. A paratopological group G is a group G with a topology such that multiplication on G is jointly continuous. A semitopological group G is a group G with a topology such that multiplication on G is separately continuous.

Recall that a space X is of (pointwise) countable type if every (point) compact subset P of X is contained in a compact subset F [subset] X that has a countable base of open neighbourhoods in X. Obviously, every space of countable type is of pointwise countable type. However, the converse is not true, even in the category of homogeneous spaces [12]. So the following result by A.V. Arhangel'skii is interesting.

Theorem 2.1. [2] Let Gbea paratopological group. If there exists a non-empty compact subset of G of countable character in G, then G is a space of countable type.

From Theorem 2.1 we know that a paratopological group of pointwise countable type is a space of countable type.

Let O be a family of open subsets of a space X and F be a subset of X. O is said to be an outer base of F in X if for each x [member of] F and each neighbourhood U of x in X there exists an element V of O such that x [member of] V [subset] U.

Recall that a space X is called Ohio complete [11], if in every compactification bX of X there exists a [G.sub.[delta]]-subset Z such that X [subset] Z and every y [member of] Z \ X is separated from X by a [G.sub.[delta]]-subset of Z. By [11] all p-spaces and all Lindelof spaces are Ohio complete.

3 Remainders of paratopological groups

To extend Arhangel'skii's result mentioned in the introduction to paratopological groups, we first prove a lemma.

Lemma 3.1. Let G be a homogeneous space with a compactification bG such that the remainder bG \ G is Ohio complete. Then either bG \ G is (Cech-complete or G is of pointwise countable type.

Proof We consider two cases.

Case 1: G is locally compact. Then G is of countable type and bG G is compact.

Case 2: G is non-locally compact. Then G is nowhere locally compact, since G is a homogeneous space. It follows that the remainder Y = bG \ G is dense in bG. Hence, bG is also a compactification of bG \ G. Since bG \ G is Ohio complete, we can fix a [G.sub.[delta]]-subset Z of bG such that Y [subset] Z and every y [member of] Z \ Y can be separated from Y by a [G.sub.[delta]]-subset of Z.

If Z \ Y is empty, then Y = Z is a [G.sub.[delta]]-subset Z of bG which implies that bG \ G is (Cech-complete.

If Z \ Y is not empty, one can take a point p [member of] Z \ Y and a [G.sub.[delta]]-subset P of Z such that p G P C Z \ Y. Then p [member of] P [subset] G, and P is a [G.sub.[delta]]-subset of bG since Z is a [G.sub.[delta]]-subset of bG. Since bG is compact, it follows that there exists a non-empty compact subset F [subset] P such that F has a countable base of open neighbourhoods in G. Since G [subset] is a homogeneous space, G has a cover by compact subsets with countable bases of open neighbourhoods in G. Then it follows that G is of pointwise countable type.

Lemma 3.2. [11] If X is a Lindelof p-space, then any remainder of X is a Lindelof pspace.

Theorem 3.1. Suppose that G is a non-locally compact paratopological group and that bG is a compactification of G. Then the remainder bG \ G is a p-space if and only if at least one of the following conditions holds:

(1) G is a Lindelof p-space;

(2) G is [sigma]-compact.

Proof Sufficiency: If G is a Lindelof p-space, then by Lemma 3.2, bG \ G is a Lindelof p-space. If G is [sigma]-compact, then bG \ G is (Cech-complete, which implies that bG \ G is a p-space.

Necessity. Since every p-space is Ohio complete [11], it follows from Lemma 3.1 that either bG \ G is (Cech-complete or G is of pointwise countable type. If bG \ G is (Cech-complete, then G is [sigma]-compact. If G is of pointwise countable type, then G is of countable type by Theorem 2.1. It follows that bG \ G is Lindelof, by Theorem 1.1. Since bG \ G is a Lindelof p-space, Lemma 3.2 implies that G is a Lindelof p-space.

Corollary 3.1. Suppose that G is a non-locally compact paratopological group with a compactification bG such that the remainder bG \ G is a paracompact p-space. Then G is a Lindelof p-space.

Proof By Theorem 3.1, G is a Lindelof p-space or a [sigma]-compact space. Suppose G is [sigma]-compact. The cellularity of a [sigma]-compact paratopological group is countable [7, Corollary 5.7.12]. It follows that c(G) [less than or equal to] [omega]. Since G is dense in bG, c(bG) [less than or equal to] [omega]. It follows that c(bG \ G) [less than or equal to] [omega], since bG \ G is dense in bG. Thus, bG \ G is Lindelof, since bG \ G is paracompact and c(bG \ G) [less than or equal to] [omega]. Since bG \ G is a Lindelof p-space and G is a remainder of bG \ G, G is a Lindelof p-space by Lemma 3.2.

By Corollary 3.1 we have the following result.

Corollary 3.2. Suppose that G is a non-locally compact paratopological group with a compactification bG such that the remainder bG \ G is a paracompact p-space. Then both G and bG \ G are Lindelof p-spaces.

Corollary 3.3. Let G be a paratopological group with a [G.sub.[delta]]-diagonal and let bG be a compacti cation o G. I the remainder bG \ G is a paracompact p-space, then G is metrizable.

Proo If G is locally compact, then G is a topological group since a locally compact paratopological group is a topological group [7, Proposition 2.3.11]. Since every locally compact space is a p-space, it follows that G is a paracompact p-space [7, Theorem 4.3.35]. It follows that G is metrizable since G has a [G.sub.[delta]]-diagonal [15].

If G is non-locally compact, then G is a Lindelof p-space by Corollary 3.1. Since G has a [G.sub.[delta]]-diagonal, it follows that G is separable and metrizable [15].

Corollary 3.4. Let G be a paratopological group with a [G.sub.[delta]]-diagonal and let bG be a

compacti cation o G. I the remainder bG \ G is a p-space, then G is a cosmic space. In particular, G is submetrizable.

Proof By Theorem 3.1, G is a Lindelof p-space or [sigma]-compact. If G is a Lindelof p-space with a [G.sub.[delta]]-diagonal, then G is separable and metrizable, which implies that G has a countable network. If G is [sigma]-compact, G has a countable network, since every compact subspace with a [G.sub.[delta]]-diagonal is separable and metrizable [15].

Since every semitopological group with countable [pi]-character has a [G.sub.[delta]]-diagonal [7, Corollary 5.7.5], by Corollary 3.4, we have the following result.

Corollary 3.5. Let G be a paratopological group with countable [pi]-character and let bG be a compactification of G. If the remainder bG \ G is a p-space, then G is a cosmic space.

Theorem 3.2. Let Gbea cosmic paratopological group with a compactification bG such that the remainder bG \ G is paracompact, then G is separable and metrizable.

Proof If G is locally compact, then G is a p-space. Since G is a cosmic space, G is a Lindelof space with a [G.sub.[delta]]-diagonal. Since every Lindelof p-space with a [G.sub.[delta]]-diagonal is separable and metrizable, so is G.

If G is not locally compact, then bG \ G is dense in bG and G is a remainder of bG \ G. Then we have c(G) [less than or equal to] [omega] since G has a countable network. Since G is dense in bG, c(bG) [less than or equal to] [omega]. It follows that c(bG \ G) [less than or equal to] [omega], since bG \ G is dense in bG. Thus, bG \ G is Lindelof, since it is paracompact and c(bG \ G) [less than or equal to] [omega].

Then G is of countable type, by Theorem 1.1. Therefore, there exists a compact subset K [subset] G such that K has a countable base of open neighbourhoods in G. Since K is compact and has a countable network, it follows that K is separable and metrizable [14]. By [17], a compact subspace F of a space X, such that F is separable and metrizable and has a countable base of open neighbourhoods in X, has a countable outer base in X. Therefore, K has a countable outer base in G. In particular, G has a countable local base at every point of K. Since G is homogeneous, it follows that G is first countable. Since G is a paratopological group, it has a countable base by a result of Ravsky (see [19, Proposition 2.13]). Therefore, G is separable and metrizable.

In Theorem 3.2, the condition that G has a countable network cannot be replaced by the weaker one that G has a countable [pi]--base. Indeed, the Sorgenfrey line G, as a paratopological group with a countable [pi]--base, has a compactification bG which is homeomorphic to the two arrows space, and the remainder bG \ G is Lindelof. However, G is non-metrizable.

In [10], Arhangel'skii proved that if a non-locally compact topological group has a compactification bG such that the remainder bG G has a [G.sub.[delta]]-diagonal, then both G and bG are separable and metrizable. However, the conclusion is false in the category of paratopological groups [10]. By Corollary 3.2, we have the following result.

Theorem 3.3. Let Gbea non-locally compact paratopological group with a compactification bG such that the remainder bG \ G is metrizable. Then both G and bG are separable and metrizable.

Proof Since bG \ G is metrizable, it is a paracompact p-space. By Corollary 3.2, both G and bG \ G are Lindelof p-spaces. Since bG \ G is metrizable and Lindelof, it follows that bG \ G is separable and metrizable.

Fix a countable base B of bG \ G. For each B [member of] B, take an open subset VB of bG such that [V.sub.B] [intersection] (bG \ G) = B. Put [O.sub.B] = [V.sub.B] [intersection] G, for each B [member of] B. Since both bG \ G and G are dense in bG, it follows that {[O.sub.B] : B [member of] B} is a countable [pi]-base of G. Since a paratopological group with countable [pi]-character has a [G.sub.[delta]]-diagonal [7, Corollary 5.7.5], G has a [G.sub.[delta]]-diagonal. Therefore, G is separable and metrizable, since G is a Lindelof p-space with a [G.sub.[delta]]-diagonal [15].

Since both G and bG \ G are separable and metrizable, it follows that bG has a countable network. Therefore, bG is separable and metrizable.

In [4] C. Liu studied local properties of remainders of a topological group and proved that if a non-locally compact topological group has a compactification bG such that bG \ G has a local [G.sub.[delta]]-diagonal, then both G and bG are separable and metrizable. For non-locally compact paratopological groups with locally metrizable remainders, we can show the following result which complements Theorem 3.3.

Theorem 3.4. Suppose G is a non-locally compact paratopological group with a compactification bG such that the remainder bG \ G is locally metrizable. Then both G and bG are separable and metrizable.

Proof Fix a point y G [member of] bG \ G and two open neighbourhoods [V.sub.y] and [W.sub.y] of y in bG \ G such that [W.sub.y] is metrizable and the closure of [V.sub.y] in bG \ G is contained in Wy. We denote by [U.sub.y] the closure of [V.sub.y] in bG G. Obviously, [U.sub.y] is metrizable.

Claim 1: [U.sub.y] is not countably compact. Suppose to the contrary that [U.sub.y] is countably compact. Then [U.sub.y] is is compact, since [U.sub.y] is metrizable. Hence, [U.sub.y] is closed in bG. On the other hand, there exists an open neighbourhood U of y in bG such that U [intersection] (bG \ G) = [V.sub.y]. Since G is non-locally compact and homogeneous, G is nowhere locally compact, which implies that bG \ G is dense in bG. Therefore, [V.sub.y] is dense in U. Thus, the closure of U in bG coincides with the closure of [V.sub.y] in bG. It follows that the closure of U in bG coincides with [U.sub.y]. This contradicts the fact that U [intersection] G [not equal to] [[empty set].

Claim 2: G has a [G.sub.[delta]]-diagonal. Since [U.sub.y] is not countably compact, there exists an infinite closed discrete countable subset F of bG \ G contained in [U.sub.y]. Since bG is compact, there exists a point c in G such that c [member of] [[bar.F].sub.bG]. Since [W.sub.y] is an open subset of bG \ G such that that [W.sub.y] is metrizable and [U.sub.y] [subset] [W.sub.y], it follows that bG \ G has countable character at each point of F. Then bG has countable character at each point of F, since bG \ G is dense in bG. For each y [member of] F, take a countable local base [O.sub.y] of y in bG, and put [B.sub.y] = {V [intersection] G : V [member of] [O.sub.y]}. Then [[union].sub.y[member of]F] [B.sub.y] is a countable n-base of c in G. Since G is homogeneous, G has countable [pi]-character. It follows that G has a [G.sub.[delta]]-diagonal [7, Corollary 5.7.5].

Let K be the closure of [U.sub.y] in bG. Obviously, K \ [U.sub.y] is a non-empty subset of G, and the interior of K \ [U.sub.y] in G is also not empty. Since [U.sub.y] is metrizable, it is Ohio complete [11]. Therefore, there exists a [G.sub.[delta]]-subset H of K such that [U.sub.y] [subset] H and every x [member of] H \ [U.sub.y] is separated from [U.sub.y] by a [G.sub.[delta]]-subset of H.

Now we show that both G and bG are separable and metrizable. For this purpose, we consider two cases.

Case 1: H \ [U.sub.y] = [pi], i.e. H = [U.sub.y]. Then K [U.sub.y] is [sigma]-compact. Since K \ [U.sub.y] is contained in G, K [U.sub.y] has a [G.sub.[delta]]-diagonal. Thus, K \ [U.sub.y] has a countable network, which implies that c(K \ [U.sub.y]) [less than or equal to] [omega]. Therefore, c(K) [less than or equal to] [omega], since K \ [U.sub.y] is dense in K. It follows from the density of [U.sub.y] in K that c([U.sub.y]) [less than or equal to] [omega]. Therefore, [U.sub.y] is separable and metrizable. Since both [U.sub.y] and K \ [U.sub.y] have countable networks, it follows that K has a countable network. Thus, K is separable and metrizable. Then K [U.sub.y] is separable and metrizable. Since G is homogeneous, it follows that G is locally separable and locally metrizable.

We claim that G is of countable type. Take an arbitrary compact subset C of G. For every x [member of] C, fix an open neighbourhood [O.sub.x] of x in G such that [O.sub.x] is separable and metrizable. Then there exists a finite subset A of G such that C C { Ox : x [member of] A}. It follows that C has a countable outer base in G. Then it is easy to see that C has a countable character in G. Thus, G is of countable type.

By Theorem 1.1, bG \ G is Lindelof. Hence bG \ G is locally separable since bG \ G is Lindelof and locally metrizable. Since bG \ G is locally separable and locally metrizable, it follows that bG \ G is separable and metrizable. Therefore, bG \ G is a Lindelof p-space. Then G is a Lindelof p-space by Corollary 3.2. Since G has a [G.sub.[delta]]-diagonal, it follows that G is separable and metrizable. Since both bG \ G and G are separable and metrizable, one can conclude that bG is separable and metrizable.

Case 2: H \ [U.sub.y] [not equal to] [[empty set]. Let O be the interior of K \ [U.sub.y] in G. Then O is dense in K. Since G is dense in bG, it follows that O [subset or equal to] [Int.sub.bG]K. We have the following two subcases.

Subcase (a): H n O = 0. Then O C K \ H, which implies that K \ H is dense in K. Since K \ H is [sigma]-compact and has a [G.sub.[delta]]-diagonal, it follows that K \ H has a countable network. Thus c(K \ H) [less than or equal to] [omega]. Since both K \ H and [U.sub.y] are dense in K, one can conclude that c([U.sub.y]) [less than or equal to] [omega]. Therefore, [U.sub.y] is separable and metrizable. Then K \ [U.sub.y] is a Lindelof p-space. It follows from the fact that K \ [U.sub.y] has a [G.sub.[delta]]-diagonal that K \ [U.sub.y] is separable and metrizable, which implies that G is locally separable and locally metrizable. As in Case 1, we come to the conclusion that both G and bG are separable and metrizable.

Subcase (b): H [intersection] O [not equal to] [[empty set]. Fix a point x [member of] H [intersection] O, then there is a [G.sub.[delta]]-subset P of H such that x [member of] P [subset] H [U.sub.y]. Since H is a [G.sub.[delta]]-subset of K, it follows that P is a [G.sub.[delta]]-subset of K. Let {Pn : n [member of] [omega]} be a sequence of open subsets of K such that P = [intersection]{[P.sub.n] : n [member of] [omega]}. Take a sequence {[W.sub.n] : n [member of] [omega]} of open neighbourhoods of x in bG such that [W.sub.0] [subset] K and [W.sub.n + 1] C [W.sub.n] [intersection] [P.sub.n]. It is easy to see that { Wn : n [member of] [omega]} is a local base of the compact set [intersection]{[W.sub.n] : n [member of] [omega]} in bG. Obviously, [intersection]{[W.sub.n] : n [member of] [omega]} is contained in G and has countable character in G. Since G is a paratopological group, it follows that G is of countable type, by Theorem 2.1. Therefore, bG \ G is Lindelof. Since bG \ G is locally metrizable, it is locally separable. It follows that bG \ G is separable and metrizable. As in Case 1, this implies that both G and bG are separable and metrizable.

Next we consider semitopological groups with remainders of countable [pi]-character.

Theorem 3.5. Suppose G is a non-locally compact separable semitopological group with a compactification bG such that the remainder bG \ G has countable [pi]-character. Then either bG \ G is countably compact or G has a countable n-base.

ProofAssume that bG \ G is not countably compact. Then there exists a countable infinite closed discrete subset F of bG \ G. Since bG is compact, there exists a point

p of G such that p [member of] [[bar.F].sub.bG]. Since G is non-locally compact, bG \ G is dense in bG. Then it follows from the fact that bG \ G has countable [pi]-character that each point of bG \ G has a countable [pi]-base in bG. For each y [member of] F, take a countable [pi]-base [O.sub.y] of y in bG, and put [B.sub.y] = {V [intersection] G : V [member of] [O.sub.y]}. Then [[union].sub.y[member of]F] By is a countable [pi]-base of p in G.

Since G is homogeneous, G has a countable [pi]-base [B.sub.e] at the identity e. Take a countable subset L of G such that L is dense in G. Put B = {xU : x [member of] L, U [member of] [B.sub.e]}. We claim that B is a countable [pi]-base of G.

Indeed, for each point a of G and an open neighbourhood W of a, there exists a point x of L such that x [member of] W. Since G is a semitopological group, there exists a neighbourhood V of e such that xV [subset] W. Since [B.sub.e] is a [pi]-base of G at e, we can find an element U G Be such that U [subset] V. Then xU [member of] B and xU [subset] W. Therefore, B is a countable [pi]-base of G.

Theorem 3.6. Suppose G is a non-locally compact cosmic semitopological group with a compactification bG such that the remainder bG \ G has countable [pi]-character. Then either bG \ G is (Cech-complete or G has a countable [pi]-base.

Proof Let N be a countable closed network of G. Denote by 7 the family of all compact elements of N. We consider two cases.

Case 1: ([union] [lambda] = G. It follows that G is [sigma]-compact, which implies that bG \ G is CCech-complete.

Case 2: G \ [union] [lambda] = 0. Fix a point a [member of] G [union] [lambda] and put B = {P [member of] N : a [member of]P}. Then [[bar.P].sub.bG] is a countable network of G at a, and none element of B is compact. Therefore, [[bar.P].sub.bG] n (bG \ G) [not equal to] [empty set], for each P G 0. Fix a point [y.sub.P] [member of] [[bar.P].sub.bG] [intersection] (bG \ G) for each P [member of] [beta], and put A = {[y.sub.p] : P [member of] [beta]}. Then A is countable and a [member of] [[bar.A].sub.bG]. Since bG \ G is dense in bG and has countable [pi]-character, it follows that bG has countable [pi]-character at every point of bG \ G. For each [y.sub.P] [member of] A, take a countable [pi]-base [O.sub.p] of [y.sub.P] in bG, and put [B.sub.p] = {V [intersection] G : V G [member of] [O.sub.p]}. Obviously, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [B.sub.p] is a countable n-base of a in G. Since G is homogeneous, G has a countable [pi]-base [B.sub.e] at the identity e. Since G has a countable network, it is separable. Let S be a countable subset of G which is dense in G, and put V = {sA : s [member of] S, A [member of] [B.sub.e]}. Then it follows from the proof of Theorem 3.5 that V is a [pi]-base of G.

Recall that a Tychonoff space X is said to be weakly pseudocompact if there exists a Hausdorff compactification bX such that X is [G.sub.[delta]]-dense in bX, that is, every non-empty [G.sub.[delta]]-set in bX intersects X.

Theorem 3.7. Suppose G is a weakly pseudocompact semitopological group with a compactification bG such that the remainder bG \ G has countable [pi]-character. Then either bG \ G is countably compact or G is a topological group metrizable by a complete metric.

ProofSuppose that bG \ G is not countably compact. Then bG \ G is not compact and, hence, G is not locally compact. As in the proof of Theorem 3.5, we see that G has countable [pi]-character. Then G has a [G.sub.[delta]]-diagonal. However, every weakly pseudocompact Tychonoff space X with a [G.sub.[delta]]-diagonal is Cech-complete [7, Proposition 5.7.19]. Further, every CCech-complete semitopological group is a topological group [13]. Then G is a topological group with countable [pi]-character, which implies that G is metrizable [16]. Since G is CCech-complete, it follows that G is a completely metrizable topological group.

Since every pseudocompact space is weakly pseudocompact and pseudocompact metrizable space is compact, the following result follows from Theorem 3.7.

Corollary 3.6. Suppose that G is a pseudocompact and non-compact semitopological group and bG is a compactification of G. If the remainder bG \ G has countable [pi]-character, then bG \ G is countably compact.

Acknowledgment

The authors would like to thank the referee for his/her valuable comments and suggestions to improve the paper.

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Institute of Mathematics, Nanjing Normal University, Nanjing 210046, China and Department of Mathematics, Shandong Agricultural University, Taian 271018, China E-mail address:weihe@njnu.edu.cn

* Project supported by NSFC (11171156)

Received by the editors in February 2013--In revised form in November 2013.

Communicated by E. Colebunders.

2010 Mathematics Subject Classification : 54D40, 54E35, 22A05.

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Author: | He, Hanfeng; Wei, Wang |
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Publication: | Bulletin of the Belgian Mathematical Society - Simon Stevin |

Article Type: | Formula |

Geographic Code: | 1USA |

Date: | Jul 1, 2014 |

Words: | 4984 |

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