Remainders of Semitopological Groups or Paratopological Groups
We mainly discuss the remainders of Hausdorff compactifications of paratopological groups or semitopological groups. Thus, we show that if a nonlocally compact semitopological group G has a compactification bG such that the remainder Y = bG \ G possesses a locally countable network, then G has a cou...
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| author | Lin, Fucai Liu, Chuan Xie, Li-Hong Лін, Фуцай Лю, Шуан Хє, Лі-Гонг |
| author_facet | Lin, Fucai Liu, Chuan Xie, Li-Hong Лін, Фуцай Лю, Шуан Хє, Лі-Гонг |
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| description | We mainly discuss the remainders of Hausdorff compactifications of paratopological groups or semitopological groups. Thus, we show that if a nonlocally compact semitopological group G has a compactification bG such that the remainder Y = bG \ G possesses a locally countable network, then G has a countable π -character and is also first-countable, that if G is a nonlocally compact semitopological group with locally metrizable remainder, then G and bG are separable and metrizable, that if a nonlocally compact paratopological group has a remainder with sharp base, then G and bG are separable and metrizable, and that if a nonlocally compact ℝ1-factorizable paratopological group has a remainder which is a k -semistratifiable space, then G and bG are separable and metrizable. These results improve some results obtained by C. Liu (Topology Appl., 159, 1415–1420 (2012)) and A.V. Arhangel’skїǐ and M. M. Choban (Topology Proc., 37, 33–60 (2011)). Moreover, some open questions are formulated. |
| first_indexed | 2026-03-24T02:19:41Z |
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| fulltext |
UDC 512.5
Fucai Lin (Minnan Normal Univ., China),
Chuan Liu (Ohio Univ. Zanesville Campus, USA),
Li-Hong Xie (Wuyi Univ., China)
REMAINDERS OF SEMITOPOLOGICAL GROUPS
OR PARATOPOLOGICAL GROUPS*
ЗАЛИШКОВI ЧЛЕНИ НАПIВТОПОЛОГIЧНИХ ГРУП
АБО ПАРАТОПОЛОГIЧНИХ ГРУП
We mainly discuss the remainders of Hausdorff compactifications of paratopological groups or semitopological groups.
Thus, we show that if a nonlocally compact semitopological group G has a compactification bG such that the remainder
Y = bG \ G possesses a locally countable network, then G has a countable π-character and is also first-countable, that
if G is a nonlocally compact semitopological group with locally metrizable remainder, then G and bG are separable
and metrizable, that if a nonlocally compact paratopological group has a remainder with sharp base, then G and bG are
separable and metrizable, and that if a nonlocally compact R1-factorizable paratopological group has a remainder which is
a k-semistratifiable space, then G and bG are separable and metrizable. These results improve some results obtained by
C. Liu (Topology and Appl. – 2012. – 159. – P. 1415 – 1420) and A. V. Arhangel’skiı̌, M. M. Choban (Topology Proc. –
2011. – 37. – P. 33 – 60). Moreover, some open questions are posed.
У данiй статтi, в основному, розглядаються залишковi члени хаусдорфових компактифiкацiй паратопологiчних груп
або напiвтопологiчних груп. Tак, показано, що у випадку, коли нелокально компактна напiвтопологiчна група G має
компактифiкацiю bG таку, що залишковий член Y = bG \G має локально злiченну мережу, група G має злiченний
π-характер, а також є першозлiченною. Також доведено, що для нелокально компактної напiвтопологiчної групи з
локально метризовним залишковим членом групи G i bG є сепарабельними i метризовними. Крiм того, якщо нело-
кально компактна паратопологiчна група має залишковий член з точною базою, то групи G i bG є сепарабельними
i метризовними, а якщо нелокально компактна R1-факторизовна паратопологiчна група має залишковий член, який
є простором, що допускає k-напiвспрямлення, то групи G i bG є також сепарабельними i метризовними. Наведенi
результати покращують деякi результати, отриманi C. Liu (Topology and Appl. – 2012. – 159. – P. 1415 – 1420) i
A. V. Arhangel’skiı̌, M. M. Choban (Topology Proc. – 2011. – 37. – P. 33 – 60). Крiм того, сформульовано деякi вiдкритi
питання.
1. Introduction. Throughout this paper, all spaces are assumed to be Tychonoff. Denote the set
of positive natural numbers by N. We refer the reader to [4, 12] for notations and terminology not
explicitly given here.
A semitopological group G is a group G with a topology such that the product map of G × G
into G is separately continuous. If G is a semitopological group and the inverse map of G onto itself
associating x−1 with arbitrary x ∈ G is continuous, then G is called a quasitopological group. A
paratopological group G is a group G with a topology such that the product maps of G×G into G
is jointly continuous. If G is a paratopological group and the inverse map of G onto itself associating
x−1 with arbitrary x ∈ G is continuous, then G is called a topological group. However, there exists
a paratopological group which is not a topological group; Sorgenfrey line [12] (Example 1.2.2)
is such an example. Paratopological groups were discussed and many results have been obtained
[4, 5, 7, 17 – 20].
Recall that a space X is of countable type if every compact subspace F of X is contained in a
compact subspace K ⊂ X with a countable base of open neighborhoods in X.
* Supported by the NSFC (No. 11201414), the Natural Science Foundation of Fujian Province (No. 2012J05013) of
China and Training Programme Foundation for Excellent Youth Researching Talents of Fujian’s Universities (JA13190).
c© FUCAI LIN, CHUAN LIU, LI-HONG XIE, 2014
500 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
REMAINDERS OF SEMITOPOLOGICAL GROUPS OR PARATOPOLOGICAL GROUPS 501
By a remainder of a space X we understand the subspace bX \ X of a Hausdorff compactifi-
cation bX of X. Remainders in compactifications of topological spaces have been studied by some
topologists in the last few years. A famous classical result in this study is the following theorem of
M. Henriksen and J. Isbell’s [15]:
A space X is of countable type if and only if the remainder in any (in some) compactification
of X is Lindelöf.
2. Paratopological groups with locally metrizable remainders. In this section we shall prove
that if a nonlocally compact semitopological group with a remainder which is locally metrizable or
has locally a countable network then G and bG are separable and metrizable.
First, we give some technical lemmas.
Lemma 2.1 [7]. Suppose that X is a regular space with a countable network1 S. Then X =
= Y ∪ Z, where Y is a separable metrizable, and Z has a countable network P such that every
element of P is nowhere dense in X.
The following lemma maybe was proved somewhere.
Lemma 2.2. Let F be a compact subset of a space X and have a countable base {Un} with
Un+1 ⊂ Un in X, and let H = ∩nVn (Vn+1 ⊂ Vn and each Vn is open in F ) is a compact Gδ-set of
F. For n ∈ N, let Wn be an open set in X such that Vn = Wn ∩ F, Wn ⊂ Un, Wn+1 ⊂ Wn, then
{Wn} is a countable base at H in X.
Proof. H = ∩nWn = ∩nWn. Suppose that {Wn} is not a countable base at H, then there is
an open subset U of X such that H ⊂ U and Wn \ U 6= ∅ for every n. By induction, choose
xn ∈ Wn \ U with xi 6= xj if i 6= j. Since xn ∈ Un for each n ∈ N, then {xn} has a cluster point
x ∈ F. Therefore, we have x ∈ Wn for each n, then x ∈ H ⊂ U, and hence U contains infinitely
many x′ns, which is a contradiction.
Lemma 2.3 [22]. Let X be a Lindelöf space with locally a Gδ-diagonal 2. Then X has a Gδ-
diagonal.
Recall that a family U of nonempty open sets of a space X is called a π-base if for each nonempty
open set V of X, there exists an U ∈ U such that U ⊂ V. The π-character of x in X is defined
by πχ(x,X) = min{|U| : U is a local π-base at x in X}. The π-character of X is defined by
πχ(X) = sup{πχ(x,X) : x ∈ X}.
Lemma 2.4 [2]. If X is a Lindelöf p-space, then any remainder of X is a Lindelöf p-space.
Theorem 2.1. If a nonlocally compact semitopological group G has a Hausdorff compactifica-
tion bG such that the remainder Y = bG\G has locally a countable network, then G has a countable
π-character and is also first-countable.
Proof. Since Y has locally a countable network, there exists an open subset U in Y such that
U
Y
has a countable network. Let V be an open subset of bG such that V ∩ Y = U. Since G is not
locally compact semitopological group, the remainder Y is dense in bG. Therefore, V
bG
= U
bG
. By
Lemma 2.1, we have U = X1 ∪ X2, where X1 is a separable metrizable subspace, and X2 has a
countable network P such that each element of P is nowhere dense in U.
Case 1: X1 is dense in U
Y
.
1Let P be a family of subsets of a space X. The family is called a network for X if, for each x ∈ U with U open in
X, there exists a P ∈ P such that x ∈ P ⊂ U.
2A space X has a Gδ-diagonal if there exists a sequence {Gn}n of open covers of X such that, for each point x ∈ X,
we have
⋂
n∈N st(x,Gn) = {x}.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
502 FUCAI LIN, CHUAN LIU, LI-HONG XIE
Since U is dense U
bG
, X1 is dense in U
bG
. Then U
bG
has a countable π-base since X1 has a
countable π-base. Therefore, V ∩G has a countable π-base, and thus G has a countable π-character.
Case 2: X1 is not dense in U
Y
.
Put W = U
bG \X1
bG
. Then W is a nonempty open subspace of U
bG
. For an arbitrary P ∈ P, let
FP = P
bG
. Since each P is nowhere dense in U, each FP is nowhere dense in U
bG
, and therefore,
each WP = W \ FP is a dense open subspace of W. Obvious, U
bG
is compact, and thus it follows
that the subspace H =
⋂
{WP : P ∈ P} of W is a Čech-complete dense subspace in W. Moreover, it
is easy to see that (V ∩G) \X1
bG 6= ∅. It follows by a standard argument that G has a dense Čech-
complete subspace, or see the proof of [6] (Theorem 1.2). Then G is a Čech-complete topological
group [7] (Corollary 5.4). Since Y has locally a countable network, G is separable and metrizable
[22]. Then G is a Lindelöf p-space, and thus, by Lemma 2.4 Y is a Lindelöf p-space. Since Y is
a Lindelöf space with locally a Gδ-diagonal, Y has a Gδ-diagonal by Lemma 2.3, and hence Y is
separable and metrizable.
Since U
Y
is Lindelöf, V
bG \UY is of countable type, and it follows from the homogeneity of G
and Lemma 2.2 that G is of countable type. Moreover, since every Tychonoff semitopological group
with a countable π-character has a Gδ-diagonal [4] (Corollary 5.7.5), G has a Gδ-diagonal. Hence G
is first-countable.
Next we shall prove that if a nonlocally compact semitopological group G has a Hausdorff
compactification bG such that the remainder bG\G is locally metrizable then G and bG are separable
and metrizable.
Lemma 2.5 [17]. Let G be a nonlocally compact semitopological group. If the remainder Y =
= bG \G is metrizable, then G and bG are separable and metrizable.
Lemma 2.6. Let X be a space with a σ-locally countable base. Then X is of countable type.
Proof. Let K be an arbitrary compact subset of X. For each x ∈ K, there exists open neigh-
borhoods Vx and Wx of x in X such that Wx ⊂ Vx and the subspace Vx has a σ-locally countable
base. Then the family of the open subsets {Wx : x ∈ K} is an open covering for K, and it fol-
lows from the compactness of K that there exist finite set {xi : 1 ≤ i ≤ n0} ⊂ K such that
K ⊂
⋃
{Wxi : 1 ≤ i ≤ n0}. For each 1 ≤ i ≤ n0, let Ki = F ∩Wx. Then each Ki is compact
and K =
⋃
1≤i≤n0
Ki. For each 1 ≤ i ≤ n0, the subspace Vxi has a σ-locally countable base
Bi =
⋃
n∈NBin, where each Bin is locally countable in Vxi , and then, for each n ∈ N the family
Din = {B ∩Ki 6= ∅ : B ∈ Bin} is countable by the compactness of Ki. Let B =
⋃
1≤i≤n0,n∈NDin.
Obviously, B is countable and each element of B is also open in X since each Vxi is open in X. Let
K =
{⋃
C : K ⊂
⋃
C and C is a finite subfamily of B
}
.
Then K is countable. Next we shall show that K is a countable base for K.
Fix arbitrary K ⊂ U with U open in X. For each x ∈ K, then there exists 1 ≤ i ≤ n0 such that
x ∈ Vxi , and thus there exists an open set Bx such that x ∈ Bx ⊂ U and Bx ∈ Din for some n. Then
{Bx : x ∈ K} is an open covering for K, and thus there is a finite subfamily K1 ⊂ {Bx : x ∈ K}
such that K ⊂
⋃
K1. Obviously,
⋃
K1 ∈ K. Therefore, K is a countable base for K.
Theorem 2.2. If a nonlocally compact semitopological group G has a compactification bG such
that the remainder Y = bG \G is locally a Σ-space with a σ-locally countable base, then G and bG
are separable and metrizable.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
REMAINDERS OF SEMITOPOLOGICAL GROUPS OR PARATOPOLOGICAL GROUPS 503
Proof. We firstly claim that Y is nowhere locally countably compact. Indeed, suppose that there
exists a ∈ Y such that a has a neighborhood U(a) in Y with U(a)
Y
countably compact. Since Y is
locally Σ-space with a σ-locally countable base, we may assume that U(a)
Y
is a Σ-subspace with
a σ-locally countable base. U(a)
Y
is compact metrizable [11]. Then U(a)
bG
= U(a)
Y ⊂ Y. Let U
be an open subset of bG such that U(a) = U ∩ Y. We have G, Y are dense in bG since G is not
locally compact, and therefore, U ∩G 6= ∅ and U
bG
= U ∩ Y bG
= U(a)
bG
= U(a)
Y ⊂ Y. This is
a contradiction. Therefore, Y is nowhere locally countably compact. Then it follows by a standard
argument that G has a countable π-character, and hence G has a Gδ-diagonal by [4] (Corollary 5.7.5).
By [11] (Corollary 7.11), Y is locally developable, hence Y is local a σ-space.
Claim: There is a point y ∈ Y such that Uy ⊂ Y is separable for some open neighborhood Uy
at y.
Suppose that Y is nowhere locally separable. Since Y is locally a Σ-space with a σ-locally
countable base, there exists an open subset U of Y such that U
Y
is a Σ-space with σ-locally countable
base. Let P = ∪n∈NPn be a σ-discrete network of U, and let Fn be the set of all accumulation points
of Pn in U
bG
for each n ∈ N. Then each Fn ⊂ G is compact and ∪n∈NFn is dense in U
bG
. Since
G has a Gδ-diagonal, Fn is compact metrizable for each n ∈ N. Then G is locally separable and
c
(
U
bG ∩G
)
≤ ω. Then it follows that c(U
bG
) ≤ ω, and hence c(U
Y
) ≤ ω. By [9] (Lemma 8.1(iii)),
every locally countable open collection in U
Y
is countable, and hence U has a countable base. Thus
U is separable and metrizable, which is a contradiction.
Since Y is locally a Σ-space with a σ-locally countable base, we may assume that the Uy in claim
is a Σ-subspace with a σ-locally countable base. Let U be an open subset such that Uy = U ∩ Y.
Since G is not locally compact, Y is dense in bG. Then it is easy to see that U
bG
= Uy
bG
. Thus
Uy
bG ∩ Y is separable in Y, and hence it is separable and metrizable [11] (Theorem 7.2). Since
Uy
bG ∩ G is a remainder of Uy
bG ∩ Y, Uy
bG ∩ G is a Lindelöf p-space by Lemma 2.4, and hence
Uy
bG ∩ G is separable and metrizable since G has a Gδ-diagonal [11] (Corollary 3.20). Then G is
locally separable and metrizable since U ∩ G ⊂ Uy
bG ∩ G and G is homogeneous. Since Y has
locally a σ-locally countable base, then Y is of countable type by Lemma 2.6.
Therefore, G is Lindelöf, and thus G is separable and metrizable. Then Y is a Lindelöf p-space
by Lemma 2.4, and hence Y is locally separable metrizable since a Lindelöf developable space
are separable and metrizable [11] (Theorem 1.2). Then Y is separable and metrizable since Y is a
Lindelöf locally separable metrizable space. By Lemma 2.5, G and bG are separable and metrizable.
Corollary 2.1. Let G be a nonlocally compact paratopological group. If the remainder Y =
= bG \G is locally metrizable, then G and bG are separable and metrizable.
3. Paratopological groups with weakly developable remainders.
Lemma 3.1 [5]. Let G be a paratopological group. Then the following conditions are equiva-
lent:
(1) some remainder Y = bG \G is Ohio-complete3;
(2) every remainder Y = bG \G is Ohio-complete;
(3) G is σ-compact or G is a space of countable type.
3A space X is Ohio complete [2] if in each compactification bX of X there is a Gδ-subset Z such that X ⊂ Z and
each point y ∈ Z \X is separated from X by a Gδ-subset of Z.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
504 FUCAI LIN, CHUAN LIU, LI-HONG XIE
Lemma 3.2 [5]. Let G be a paratopological group. If there exists a nonempty compact subset
of G of countable character in G, then G is of countable type.
Theorem 3.1. Let G be a nonlocally compact paratopological group. If the remainder Y =
= bG \G satisfies the following conditions, then G and bG are separable and metrizable.
(1) Y is Ohio-complete.
(2) Y is a locally p-space with a point countable base.
Proof. Since Y is Ohio-complete, it follows from Lemma 3.1 that G is σ-compact or G is a
space of countable type.
Case 1: G is a space of countable type.
By Henriksen and Isbell’s theorem, Y is Lindelöf. Since Y is a locally p-space with a point
countable base, Y is locally metrizable since a paracompact p-space with a point-countable base
is metrizable [11] (Corollaries 3.20 and 7.11), and then G and bG are separable and metrizable by
Corollary 2.1.
Case 2: G is σ-compact.
Since G is a σ-compact paratopological group, the Souslin number c(G) of G is countable [4]
(Corollary 5.7.12). Therefore, c(bG) ≤ ω. Y is dense in bG, since G is nonlocally compact. It follows
that c(Y ) ≤ ω as well. Since Y is Čech-complete, there exists a dense subspace Z ⊂ Y such that
Z is a paracompact and Čech-complete subspace of Y by [24]. Since Z is a locally paracompact
Čech-complete subspace with a point-countable base, Z is locally metrizable [11] (Corollaries 3.20
and 7.11). Since c(Y ) ≤ ω and Z is dense for Y, c(Z) ≤ ω as well. It follows that Z is locally
separable, and hence Y is locally separable since Z is dense in Y. Then Y is locally separable space
with a point-countable base, and hence Y has locally a countable base, which implies that Y is locally
metrizable. Then G and bG are separable and metrizable by Corollary 2.1.
Corollary 3.1. Let G be a nonlocally compact paratopological group. If the remainder Y =
= bG \G satisfies one of the following conditions, then G and bG are separable and metrizable.
(1) Y is a p-space with a point-countable base.
(2) Y has a sharp base4.
Proof. (1) Since a p-space is Ohio-complete [2], it follows from Theorem 3.1 that if Y is a
p-space with a point-countable base then G and bG are separable and metrizable.
(2) Since Y has a sharp base, it follows from [8] (Theorem 3.4) that Y is a weakly developable5
space. Therefore, Y is a p-space by [8] (Theorem 2.4), and hence Y is Ohio-complete [2]. Since Y
has a sharp base, Y has a point-countable base [1]. Then Y is a p-space with a point-countable base,
and hence G and bG are separable and metrizable by (1).
Corollary 3.2. Let G be a nonlocally compact paratopological group. If the remainder Y =
= bG \ G has a uniform base (that is, a metacompact developable space), then G and bG are
separable and metrizable.
Theorem 3.2. Let G be a nonlocally compact paratopological group. If the remainder Y =
= bG \G is weakly developable and irresolvable, then G and bG are separable and metrizable.
4A sharp base B of a space X is a base of X such that, for every sequence {Bn : n ∈ N} of distinct members of B
and every x ∈
⋂
n∈NBn, the sequence {
⋂
i≤nBi : n ∈ N} is a base at x.
5A space X is called weakly developable if there exists a sequence {Gn : n ∈ N} of open covers on X such that for
every sequence {Bn ∈ Gn : n ∈ N} and every x ∈
⋂
n∈NBn, the sequence {
⋂
i≤nBi : n ∈ N} is a base at x.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
REMAINDERS OF SEMITOPOLOGICAL GROUPS OR PARATOPOLOGICAL GROUPS 505
Proof. By the proof of Theorem 3.1, it is suffice to consider the case of G is σ-compact.
Moreover, it follows from the proof of Theorem 3.1 that the Souslin number c(Y ) of Y is countable
and there exists a dense subspace Z ⊂ Y such that Z is separable and metrizable subspace of Y. Put
X1 = bG \ Z and X2 = Y \ Z.
Obvious, Z
bG
= bG, and therefore X1 is the remainder of Z. Since Z is separable and metrizable,
Z is a Lindelöf p-space, and hence X1 is a Lindelöf p-space by Lemma 2.4. Since Y is irresolvable,
we have X2
bG 6= bG, and thus X2
bG∩G 6= G. Therefore, X1 \X2
bG ⊂ G is a nonempty open subset
in X1. Since X1 is a p-space, X1 is a space of point-countable type. Take a point x0 ∈ X1 \X2
bG
.
Then there exists a compact subset F ⊂ X1 such that x0 ∈ F and F has a countable neighborhoods
base at F. By Lemma 2.2, there exists a compact subset L ⊂ X1 \X2
bG
such that x0 ∈ L ⊂ F and
L has a countable neighborhoods base at L. It follows from Lemma 3.2 that G is of countable type.
By Henriksen and Isbell’s theorem, Y is Lindelöf. Since Y is weakly developable, Y is metrizable
by [8] (Proposition 2.6), and then G and bG are separable and metrizable by Lemma 2.5.
Theorem 3.3. Let G be a nonlocally compact paratopological group which is a generalized
ordered space (that is, GO-space). If the remainder Y = bG \G is locally weakly developable, then
G and bG are separable and metrizable.
Proof. In view of proof Theorem 2.2, we have Y is nowhere locally countably compact, and
hence Y is not countably compact, Then it follows by a standard argument that G has a countable
π-character, and hence G has a Gδ-diagonal by [4] (Corollary 5.7.5). Since a GO-space with a Gδ-
diagonal is first-countable [10] (Lemma 5.1 and Proposition 5.5). Therefore, G is countable type by
Lemma 3.2. By Henriksen and Isbell’s theorem, Y is Lindelöf. Since Y is locally weakly developable,
Y is locally metrizable by [8] (Proposition 2.6), and then G and bG are separable and metrizable by
Corollary 2.1.
Corollary 3.3. Let G be a nonlocally compact paratopological group which is GO-space. If the
remainder Y = bG \G is locally developable, then G and bG are separable and metrizable.
However, the following question is still open.
Question 3.1. Let G be a nonlocally compact paratopological group. If the remainder Y =
= bG \G is developable, are G and bG separable and metrizable?
Lemma 3.3 [5]. Let G be a k-gentle paratopological group, and Y be a remainder of G. Then
Y is Lindelöf or pseudocompact.
Lemma 3.4 [5]. Let G be a k-gentle paratopological group such that some remainder of G is
Lindelöf. Then G is a topological group.
Theorem 3.4. Suppose that G is a nonlocally compact, k-gentle paratopological group, and
Y = bG \G is a remainder of G. If Y has a weakly uniform base6, then G, bG and Y are separable
and metrizable spaces.
Proof. Since Y has a weakly uniform base, Y has a Gδ-diagonal [16], and therefore, Y is
Ohio-complete. By Lemma 3.1, G is a space of countable type or G is σ-compact.
Case 1: G is a space of countable type.
By Henriksen and Isbell’s theorem, Y is Lindelöf. By Lemma 3.4, G is a topological group.
Since Y has a Gδ-diagonal, G, bG and Y are separable and metrizable spaces [3].
Case 2: G is σ-compact.
6A base B for a space X is said to be weakly uniform if for each countably infinite family U ⊂ B and for each x ∈ X,
if x ∈ U for each U ∈ U, then
⋂
U = {x}.
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506 FUCAI LIN, CHUAN LIU, LI-HONG XIE
By the proof of Theorem 3.1, we have c(Y ) ≤ ω. It follows from Lemma 3.3 that Y is Lindelöf
or pseudocompact. By the case 1, it is suffice to consider the case of pseudocompactness of Y. Let Y
be pseudocompact. Since a pseudocompact ccc space with a weakly uniform base is metrizable [23],
Y is metrizable. Then G and bG are separable and metrizable by Lemma 2.5.
However, the following question is still open.
Question 3.2 [20]. Suppose that G is a nonlocally compact, k-gentle paratopological group, and
Y = bG \G is a remainder of G. If Y has a Gδ-diagonal, are G, bG and Y separable and metrizable
spaces?
The following theorem is also a partial answer to Questions 3.1 and 3.2.
Theorem 3.5. Let G be a nonlocally compact paratopological group. If the remainder Y =
= bG \G satisfies one of the following conditions, then G and bG are separable and metrizable.
(1) Y is a meta-Lindelöf 7 developable space;
(2) G is k-gentle and Y is a meta-Lindelöf space with a Gδ-diagonal.
Proof. Since Y has a Gδ-diagonal, Y is Ohio-complete. By Lemma 3.1, G is a space of countable
type or G is σ-compact.
Case 1: G is a space of countable type.
By Henriksen and Isbell’s theorem, Y is Lindelöf.
(1) If Y is developable, then Y is metrizable [11], and hence G and bG are separable and
metrizable by Corollary 2.1.
(2) If G is k-gentle and Y is a meta-Lindelöf space with a Gδ-diagonal, then it follows from
Lemma 3.4 that G is a topological group. Since Y has a Gδ-diagonal, G, bG and Y are separable
and metrizable spaces [3].
Case 2: G is σ-compact.
By the proof of Theorem 3.1, we have c(Y ) ≤ ω, Y is Čech-complete, and there exists a
dense subspace Z ⊂ Y such that Z is a paracompact Čech-complete subspace of Y. Obvious, we
have c(Z) ≤ ω. Since a paracompact Čech-complete space with a Gδ-diagonal is metrizable [11]
(Corollaries 3.8 and 3.20), Z is metrizable. Then Z is separable since c(Z) ≤ ω, and hence Y is
separable. Since Y is meta-Lindelöf, then Y is Lindelöf. By case 1, one obtain that G and bG are
separable and metrizable.
Finally, we pose some open questions.
Question 3.3. Let G be a nonlocally compact paratopological group. If the remainder Y =
= bG \G has locally sharp base, are G and bG separable and metrizable?
Question 3.4. Let G be a nonlocally compact semitopological group. If the remainder Y =
= bG \G has a sharp base, are G and bG separable and metrizable?
Question 3.5. Let G be a nonlocally compact paratopological group which is GO-space. If the
remainder Y = bG \G has a point-countable base, are G and bG separable and metrizable?
Question 3.6. Let G be a nonlocally compact paratopological group. If the remainder Y =
= bG \G has a weakly uniform base, are G and bG separable and metrizable?
7A space X is said to be meta-Lindelöf if each open cover of X has a locally countable open refined covering.
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REMAINDERS OF SEMITOPOLOGICAL GROUPS OR PARATOPOLOGICAL GROUPS 507
4. The remainders of R1-factorizable paratopological groups. A paratopological group H is
called R1-factorizable [25] if H is a T1-space and for every continuous real-valued function f on H,
one can find a continuous homomorphism p : H → K onto a paratopological group K of countable
weight satisfying the T1 separation axiom and a continuous real-valued function g on K such that
f = g ◦ p.
Remark 4.1. In this paper, we assume that all H in the above definition are Tychonoff.
A space (X, τ) is called a k-semistratifiable space if there exists a function S : N× τ → τ c such
that:
(a) for each U ∈ τ, U =
⋃
{S(n,U) : n ∈ N};
(b) if U, V ∈ τ and U ⊂ V, then S(n,U) ⊂ S(n, V ) for each n ∈ N;
(c) for each compact K of X and open neighborhood U of K, there exists an n ∈ N such that
K ⊂ S(n,U).
Lemma 4.1 [25]. Let G be R1-factorizable paratopological group. Then ω(G) = χ(G).
By Theorem 2.1 and Lemma 4.1, it is easy to see the following theorem holds.
Theorem 4.1. Let G be a nonlocally compact R1-factorizable paratopological group. If the
remainder Y = bG\G has locally a countable network, then G and bG are separable and metrizable.
Theorem 4.2. Let G be a nonlocally compact R1-factorizable paratopological group. If the
remainder Y = bG \G is a k-semistratifiable space, then G and bG are separable and metrizable.
Proof. Since Y is a k-semistratifiable space, Y is a σ-space [11], and hence Y has a Gδ-diagonal,
and hence Y is Ohio-complete [2]. By Lemma 3.1, G is σ-compact or G is a space of countable
type.
Case 1: G is a space of countable type.
By Henriksen and Isbell’s theorem, Y is Lindelöf. Then Y is a Lindelöf σ-space, and hence Y
has a countable network by [11] (Theorem 4.4). Therefore, G is first-countable by Theorem 2.1, and
thus it follows from Lemma 4.1 that G is separable and metrizable. Then G is a Lindelöf p-space,
and hence Y is a Lindelöf p-space by Lemma 2.4. Thus Y is metrizable by [11] (Corollaries 3.8 and
3.20). Then G and bG are separable and metrizable by Lemma 2.5.
Case 2: G is σ-compact.
SinceG is a σ-compact paratopological group, Y is Čech-complete, and hence Y is first-countable
[12]. Then Y is a stratifiable space since a Fréchet k-semistratifiable space is stratifiable [14], and
hence Y is paracompact. By the proof of Theorem 3.1, we have c(Y ) ≤ ω, and thus Y is Lindelöf.
By case 1, G and bG are separable and metrizable.
Corollary 4.1. Let G be a nonlocally compact R1-factorizable paratopological group. If the
remainder Y = bG \G is an ℵ-space, then G and bG are separable and metrizable.
By [25] (Corollaries 3.10 and 3.14), we know that if paratopological groups have a countable
network or are σ-compact then they are R1-factorizable, and hence we have the following corollary.
Corollary 4.2. Let G be a nonlocally compact paratopological group, and the remainder Y =
= bG \G be a k-semistratifiable space. If G satisfies one of the following conditions, then G and bG
are separable and metrizable.
(1) G has a countable network.
(2) G is σ-compact.
However, the following question is still open.
Question 4.1. Let G be a nonlocally compact R1-factorizable paratopological group. If the re-
mainder Y = bG \G is a σ-space, then are G and bG separable and metrizable?
The following theorem is also a partial answer to Question 4.1.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
508 FUCAI LIN, CHUAN LIU, LI-HONG XIE
Theorem 4.3. Let G be a nonlocally compact R1-factorizable paratopological group. If the
remainder Y = bG \G is a meta-Lindelöf σ-space, then G and bG are separable and metrizable.
Proof. By the proof of Theorem 4.2, it is suffice to prove that Y is Lindelöf if G is σ-compact.
Let G be σ-compact. By the proof of Theorem 3.1, we have c(Y ) ≤ ω, Y is Čech-complete, and
there exists a dense subspace Z ⊂ Y such that Z is a paracompact Čech-complete subspace of
Y. Obvious, we have c(Z) ≤ ω. Since a paracompact Čech-complete space with a Gδ-diagonal is
metrizable [11] (Corollaries 3.8 and 3.20), Z is metrizable. Then Z is separable since c(Z) ≤ ω, and
hence Y is separable. Since Y is meta-Lindelöf, then Y is Lindelöf.
Question 4.2. Let G be a nonlocally compact R1-factorizable paratopological group. If the re-
mainder Y = bG \G is a locally ℵ-space, then are G and bG separable and metrizable?
Theorem 4.4 [21]. Let G be a nonlocally compact paratopological group. Then either every
remainder of G has the Baire8 property, or every remainder of G is meager9 and Lindelöf.
Theorem 4.5. Let G be a R1-factorizable paratopological group with a Gδ-diagonal. If G is
nonmetrizable or nonseparable, then the remainder Y = bG \G is Baire.
Proof. By Theorem 4.5, Y is meager and Lindelöf or Y is Baire. Suppose that Y is meager and
Lindelöf. Then G is of countable type, and thus G is first-countable since G has a Gδ-diagonal. It
follows from Lemma 4.1 that G is separable and metrizable, which is a contradiction.
Theorem 4.6. LetG be a R1-factorizable nonmetrizable or nonseparable paratopological group.
If for each point y ∈ Y = bG\G there is an open neighborhood V (y) of y such that every countably
compact subset of V (y) is metrizable and the remainder Y is of countable π-character, then Y is
Baire.
Proof. If G is locally compact, then the remainder is compact by [12] (Theorem 3.5.8), hence
it is Baire. If G is nonlocally compact, then we may use the proof of Theorem 4.4 to prove that the
remainder is Baire.
Theorem 4.7. Let G be a R1-factorizable paratopological group, and the remainder Y = bG\G
be a k-space with a locally point-countable k-network. If Y is not Baire and is of countable π-
character, then G and bG are separable and metrizable.
Proof. Since a countably compact k-space with a point-countable k-network is metrizable [13],
it follows from Theorem 4.6 that G is metrizable, and hence G is separable and metrizable by
Lemma 4.1. Then G is a Lindelöf p-space, and thus Y is a Lindelöf p-space by Lemma 2.4. Then Y
is a Lindelöf p-space with a point-countable k-network, and thus Y is metrizable by [13]. Then G
and bG are separable and metrizable by Lemma 2.5.
Corollary 4.3. Let G be a R1-factorizable paratopological group. If the remainder Y = bG \G
is not Baire space with a locally point-countable base, then G and bG are separable and metrizable.
Question 4.3. Let G be a R1-factorizable paratopological group. If the remainder Y = bG \ G
is a space with a locally point-countable base, then are G and bG separable and metrizable?
Question 4.4. Let G be a R1-factorizable paratopological group. Is the remainder Y = bG \ G
Lindelöf or pseudocompact?
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Received 18.06.12,
after revision — 27.01.13
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| id | umjimathkievua-article-2152 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:19:41Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/bb/91b4eb72a1e008f2f1333ca0300b20bb.pdf |
| spelling | umjimathkievua-article-21522019-12-05T10:25:15Z Remainders of Semitopological Groups or Paratopological Groups Залишкові члени напівтопологічних груп або паратопологічних груп Lin, Fucai Liu, Chuan Xie, Li-Hong Лін, Фуцай Лю, Шуан Хє, Лі-Гонг We mainly discuss the remainders of Hausdorff compactifications of paratopological groups or semitopological groups. Thus, we show that if a nonlocally compact semitopological group G has a compactification bG such that the remainder Y = bG \ G possesses a locally countable network, then G has a countable π -character and is also first-countable, that if G is a nonlocally compact semitopological group with locally metrizable remainder, then G and bG are separable and metrizable, that if a nonlocally compact paratopological group has a remainder with sharp base, then G and bG are separable and metrizable, and that if a nonlocally compact ℝ1-factorizable paratopological group has a remainder which is a k -semistratifiable space, then G and bG are separable and metrizable. These results improve some results obtained by C. Liu (Topology Appl., 159, 1415–1420 (2012)) and A.V. Arhangel’skїǐ and M. M. Choban (Topology Proc., 37, 33–60 (2011)). Moreover, some open questions are formulated. У даній статті, в основному, розглядаються залишковi члени хаусдорфових компактифiкацiй паратопологічних груп або напівтопологічних груп. Tак, показано, що у випадку, коли нелокально компактна напівтопологічна група $G$ має компактифікацію $bG$ таку, що залишковий член $Y = bG \backslash G$ має локально злічєнну мережу, група $G$ має злічєнний π-характер, а також є першозліченною. Також доведено, що для нелокально компактної напівтопологічної групи з локально метризовним залишковим членом групи $G$ i $bG$ є сепарабельними i метризовними. Крім того, якщо нелокально компактна паратопологічна група має залишковий член з точною базою, то групи $G$ i $bG$ є сепарабельними і метризовними, а якщо нелокально компактна $ℝ_1$ -факторизовна паратопологічна група має залишковий член, який є простором, що допускає $k$-напівспрямлення, то групи $G$ i $bG$ є також сепарабельними i метризовними. Наведені результати покращують деякі результати, отримані C. Liu (Topology and Appl. - 2012. - 159. - P. 1415-1420) i A. V. Arhangel'skii, M. M. Choban (Topology Proc. - 2011. - 37. - P. 33 - 60). Крім того, сформульовано деякі відкриті питання. Institute of Mathematics, NAS of Ukraine 2014-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2152 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 4 (2014); 500–509 Український математичний журнал; Том 66 № 4 (2014); 500–509 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2152/1306 https://umj.imath.kiev.ua/index.php/umj/article/view/2152/1307 Copyright (c) 2014 Lin Fucai; Liu Chuan; Xie Li-Hong |
| spellingShingle | Lin, Fucai Liu, Chuan Xie, Li-Hong Лін, Фуцай Лю, Шуан Хє, Лі-Гонг Remainders of Semitopological Groups or Paratopological Groups |
| title | Remainders of Semitopological Groups or Paratopological Groups |
| title_alt | Залишкові члени напівтопологічних груп або паратопологічних груп |
| title_full | Remainders of Semitopological Groups or Paratopological Groups |
| title_fullStr | Remainders of Semitopological Groups or Paratopological Groups |
| title_full_unstemmed | Remainders of Semitopological Groups or Paratopological Groups |
| title_short | Remainders of Semitopological Groups or Paratopological Groups |
| title_sort | remainders of semitopological groups or paratopological groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2152 |
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