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Topology Proceedings
Department of Mathematics & Statistics
Auburn University, Alabama 36849, USA
[email protected]
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Topology Proceedings
Vol 17, 1992
DISCRETE CHAIN CONDITIONS AND THE
B-PROPERTY
YINZHU GAO, HANZHANG QU AND SHUTANG WANG
ABSTRACT. In this paper, we introduce and study A­
Lindelof, '\-compact and D,\Ce spaces. Some character­
izations of these spaces are given. It is shown that in
T 1 spaces with the B-property, wa-Lindelofness, Wa+l­
compactness and the DwaCC coincide. Some known re­
sults in the literature are generalized.
1.
PRELIMINARIES
Throughout the paper, _by a space we mean a topological
space without any separation axioms assumed unless especial­
ly stated. Regular spaces need not be Tt . Cardinals are initial
ordinals. w = Wo denotes the first infinite ordinal and A always
denotes an infinite cardinal. A space X is said to have the
V-property [MN] (resp. B-property[Z]) if any increasing open
cover {Ua : 0' E A} of X has an open refinement (resp. in­
creasing open refinement) {Va: 0' E A} such that Va C Uo for
any
Q
E A. It is known that
(*) paracompactness --+ B-property --+ V-property --+
countable paracompactness
The above implications are not reversible (see [MN]). A fam­
ily U of subsets of X is called a point-A (resp. point-finite)
family if for every x E X,I{U E U : x E Xli ~ A (resp.
< w). A family A = {A o : 0' E A} of subsets of a space X
is hereditarily closure-preserving (shortly, HCP) if any family
{B a : Q' E A} with B o C A o for each 0: E A is closure­
preserving. Clearly, local finiteness implies HCP and HCP im­
97
98
YINZHU GAO, HANZHANG QU AND SHUTANG WANG
plies closure-preservingness. The following symbols are used:
AF(X) = {U : U is an open cover of X having no finite sub­
cover}. A~(X) = {U : U is an open cover of X having no
subcover with cardinality ~ A}. If AF(X) and A~(X).are not
empty, we put fF(X) = min{IUI : U E AF(X)} and f~(X) =
min{IUI : U E A~(X)}
2. DEFINITIONS
Definition 2.1. A space X is said to be A-Lindelof if any
open cover of X has a subcover with cardinality
~
A.
Definition 2.2. A space X is said to be A-compact if any sub­
set with cardinality A has an accumulation point.
Definition 2.3. A space X is called a DACC (discrete A chain
condition) space if any discrete family of non-empty open sets
has cardinality < A.
Clearly, w- Lindelof spaces coincide with Lindelof spaces and
a T1 space is w-compact iff it is countably compact. WI-compact
spaces are well known. DwCC is also denoted by DCCC which
was introduced in [W]. DFCC denotes that any discrete family
of non-empty open sets is finite. In the following Figure 1, the
implications are obvious and none of them is reversible.
Example 2.1. (1) It is well known that [O,Wl) is w-compact,
but not compact.
(2) [0,W a +2) is Wa+l-compact, but not wa+l-Lindelof (hence
notwa-Lindelof). In fact, let A C [0,W a +2) and IAI = Wa +l,
then {30 = supA < W a +2 since W a +2 is regular. Since [0, {30] is
compact the infinite subset A of [0, {jo] has an accumulation
point E [0, (30] which is also an accumulation point of A in
[0,W a +2). So [0,Wa +2) is Wa+l-compact. Take the open cover
U = {[a, (3) : {3 E (0,W a +2)} of [O,W a +2). For any U C U with
lUI ~ Wa+l, let e = {(3 : [0, (3) E U}, then lei ~ Wo+I and
Ao = sup e < W a +2 since W a +2 is regular. Take a A such that
Ao < A < W a +2, then for any [0, (3) E U, we have AE[O, (8). This
shows that [0,W a +2) is not wO+I-Lindelof.
e
DISCRETE CHAIN CONDITIONS AND THE 8-PROPERTY 99
compact
~
1
1
w-Lindelof
w-compact
~
WI-compact
!
!
WI-Lindelof --+
W2-compact
!
!
!
1
wo-Lindelof
!
--+
Wo+l-compact
!
wO+I-Lindelof --+ Wo+2- corn pact
1
1
T1
----+
T1
----+
T1
----+
DFCC
1
DwCC
!
DwlCC
!
T1
----+
T1
----+
!
DwoCC
!
DWo+ICC
1
Figure 1
Example 2.2. (1) Let X be a set with IXI = W o +2 and
T = {0}u{U eX: IX -UI ~ Wo+I}, then the T1 space (X, T)
is DFCC (hence DwoCC), -but is not Wo+l-cornp'act (hence not
w-compact) .
(2) Let A be a set of cardinality A and {Ns : s E S}, where
S n A = 0 and each N s C A has cardinality w, be an infinite
family satisfying that the intersection NsnNs is finite for every
pair s, s of distinct elements of Sand {Ns : s E 5} is maximal
with respect to the last property. lGenerate a topology on the
set X = A U S by the neighborhood system {B(x) : x EX},
where B(x) = {{x}} if x E A and B(x) = {{s} U (Ns - F): F
is a finite subset of N s } if x = s E S. The space X is Tychonoff
and DFCC (hence DACC), but not A-compact (cf. 3.6.I(a) of
[E]).
Example 2.3. Suppose X is a discrete space.
(1) Let IXI = w, then X is w-Lindelof (hence WI-compact,
DwCC), but not DFCC (hence not w-cornpact, not compact).
(2) Let IXI = W a+l, then X is wO+I-Lindelof (hence W o +2­
compact, DWQ+I CC), but not DwQCC (hence not WQ+I-compact,
100 YINZHU GAO, HANZHANG QU AND SHUTANG WANG
not wo-Lindelof).
3. CHARACTERIZATIONS
Definition 3.1. A space X is said to have the property (A) if
any increasing open cover U of X with lUI ~ A has a closed
refinement F with IFI ~ A.
. Clearly, for any A, the V-property (hence the B-property)
implies the property (A). A space is countably metacompact[MN]
if every countable open cover of it has a point-finite open refine­
ment. Clearly, countable paracompactness implies countable
metacompactness. Form the following proposition, we can see
that in a normal space countable paracompactness and count­
able metacompactness are equivalent.
Proposition 3.1. A space X is countable metacompact iff ev­
ery increasing countable open cover {Ui : i < w} of X has
an increasing countable closed refinement {Fi : i < w} such
that Fi C Ui for any i < w. Hence countable metacompactness
implies" the property (w).
Proof: Necessity: Let V be a point-finite open refinement of
the increasing open cover U = {Ui : i < w}. Put Fi = {x EX:
st(x, V) CUi}' then {Fi : i < w} is the desired refinement of
U.
Sufficiency: Let U = {Ui : i < w} be a countable open cover
of X. Put Vi = Uj<iUj, then the increasing open cover {Vi : i ~
w} has an increasing closed refinement {Fi : i < w} such that
Fi c Vi for any i < w. Put Wo = Vo, Wi = Vi - Fi - 1 , i ~ 1,
then {Wi n Uj : j ~ i, i < w} is a point-finite open refinement
ofU.
Lemma 3.1. (1) fF(X) is regular. (2) If the space X has the
property (A), f~(X) is regular.
Proof: (1) Suppose fF(X) = K > cfK and U = {Ua : 0 <
K,} E AF(X) such that lUI = fF(X). Let f : cfK, --+ K be
an increasing cofinal mapping and for each 0 < cfK" Wo =
Ufj<J(o)Ufj. Then W = {Wo : 0 < cfK,} is an increasing open
DISCRETE CHAIN CONDITIONS AND THE B-PROPERTY 101
cover of X with IWI ~ cfK. Thus W has a finite (increasing)
subcover W. So there is a Woo E W such that X = Woo =
Up<f(oo)Up . Since f( (0) < K, {LJ:B : (3 < f( oo)} has a finite
subcover U CU. This contradicts the choice of U.
(2) Suppose l~(X) = K > cfK and U = {Ua : (} < K} E
A~(X) such that lUI = l;\(X). Then the increasing open cover
W,. as defined in case (1), has a (increasing) subcover W with
IWI ~ A. By the property (A), W; has a closed refinement :F
with I:FI ~ A. For each F E :F, there is a WaF E W such
that F C WOF = U/3<f(OF)U/3. Therefore {X - F} U {U{3 :
(3 < f( OF)} is an open cover of 4X with cardinality < K. It
follows that for each F there is some UF C U that covers F
and IUFI ~ A. Hence U{UF : F E :F} is a cover of X with
cardinality :5 A, contradicting the choice of U.
It is interesting that a space X is compact iff any increasing
open cover of X has a finite subcover [Al. For A-Lindelof spaces,
we have
Theorem 3.1. Let the space X ha've the property (A). Then X
is A-Lindelof iff any increasing ope'n cover of X has a subcover
with cardinality ~ A.
Proof: Necessity is obvious. Sufficiency: Suppose that X is
not A-Lindelof. Take U E A;\(X) such that lUI = l;\(X) =
K,. Let U = {Uo : 0 < K,} and 'Wo = U/3<oU{3. Then the
increasing open cover W = {Wo : Q < K,} of X has a subcover
W = {W"'p : f3 < Kt}, where k1 :::; A. Put U = Up<ICl {Ue E
U : ~ < 0/3}. By Lemma 3.1, K is regular. Since for every
(3 < K,1,QI3 < I\, and KI ~ A < K, we have lUI < k. According
to the definition of /\', U has a subcover V with IVI $ A. This
contradicts the choice of U. Noticing the case A = wand
Proposition 3.1, we have
Corollary 3.1. A countably metacompact space (hence a count­
ably paracompact space) is Lindelof iff every increasing open
cover of it has a countable subcover.
102 YINZHU GAO, HANZHANG QU AND SHUTANG WANG
Remark 3.1. We do not know whether the condition that
X· has. the property (A) in Lemma 3.1.(hence in Theorem 3.1)
can be removed. For the case A = w, M. E. Rudin has a
conjecture that there is a normal, non-Lindelof space, every
increasing open cover of which has a countable subcover (see
[MR] , Chapter 10). By Corollary 3.1, if such a space exists,
it. must be a Dowker space ( i.e. a normal but not countably
paracompact space).
Theorem 3.2. For a T 1 space X, the following are equlva­
lent:
(1) X is A-compact.
(2) Any discrete closed subspace Y of X has cardinality <
A.
(3) Any discrete family U of non-empty subsets of X has
cardinality < A.
(4) Any discretefamilyU of non-empty closed subsets of X
has cardinality < A.
(5) Any irreducible open cover U of X has cardinality < A.
(6) Any point-A, irreducible open cover U of X has cardi­
nality < A.
Proof: (1)~(2), (2) ~ (3) and (3) ~ (4) are obvious. (4)
(5): Let U = {Uo : 0 E A} be an irreducible open cover.
Since U is irreducible, for every 0 E A, we can take X E
Uo - U~~oU~. Put M = {x a : 0 E A}. For every x E X,
there is an 00 E A such that x E UOIO and UOI n M = {x OlO } .
-+
Q
= {{ x a } : 0 E A} is a discrete family of closed sets. By
(4), lUI = IK:I < A. (5) -+ (6) is obvious. (6) -+ (1): Suppose
X is not A-compact, then there is an M C X with IMI = A
So K,
such that M has no accumulation point. So M is closed. For
every x EM, there is an open set Ux such that x E Ux and
Ux n M = {x}. Then U = {Ux : x E M} U {X - M} (U =
{Ux : x E M} if U{Ux : x E M} = X) is an irreducible open
cover satisfying for any x E X, I{U E U : x E U}I ~ A and
lUI = A. This contradicts (6).
DISCRETE CHAIN CONDITIONS AND THE B-PROPERTY 103
Remark 3.2. In this remark, spaces are T1 • So a space is
w-compact. iff it is countably compact. Since any point-finite
cover of a space has an irreducible subcover[E], by the equiva­
lence of (1) and (5) of Theorem 3.2, we obtain
(1) A metacompact space[MN] is Wo+l-compact (resp. count­
ably compact) iff it is wo-Lindelof (resp. compact).
By the equivalence of (1) and (4) of Theorem 3.2, we obtain
(2) A subparacompact space[MN] is Wo+l-compact iff it is W o ­
Lindelof.
Noticing that if for any discrete family {{x(3} : {3 < Wo +l},
there is a discrete family {G(3 : (3 < WQ+l} of open sets such
that G{3 ::J {X{3}, then the DwQCC implies WQ+l-compactness,
we obtain
(3) A collectionwise normal spac.e is WQ+I-compact iff it sat­
isfies the DwOtCC.
We can prove the following (4) similar to Lemma 1.2 in [T].
Actually, Lemma 1.2 in [T] is the case a = 0 in (4).
(4) Let X be a space and K, a cover of X by wQ-compact
sets. If X has a (mod K,)-~et F such that IFI ~ w o , then X is
Wo+l-compact.
It is easy to show that in a pseudonormal space [H) for any
discrete countable family {{ Xi} : i < w}, there is a discrete
family {G i : i < w} of open sets such that Xi E Gi for any
i < w. So we have
(5) A pseudonormal space is countably compact iff it satisfies
the DFCC.
There is a Tychonoff DFCC space which is not countably
compact (see Example 2.2 (2)).
Theorem 3.3. For a regular space' X, the following are equiv­
alent:
(1) X satisfies the DACC (resp. DFCC).
(2) Any locally finite family U of open sets of X has cardi­
nality :5 A (resp. < w).
(3) Any locally finite open cover U of X has cardinality :5 A
(resp. < w).
104 YINZHU GAO, HANZHANG QU AND SHUTANG WANG
(4) Any locally finite i1Teducible open coverU of X has car­
dinality ~,\ (resp. < w).
(5) Any point-'\ (resp. point-finite), HCP family U of open
sets of X has cardinality ~ ,\ (resp. < w).
(6) Any point-'\ (resp. point-finite), HCP open cover U
of X has cardinality ~ ,\ (resp. < w).
(7) Any point-'\ (resp. point-finite), HCP irreducible open
cover U of X has cardinality ~ ,\ (resp. < w).
Proof: (1) --+ (5): Let X be well-ordered and x E X. Put
U(x) = {U E U : x is the first element in U}. Since U is a
point-'\ (resp. point-finite) family, U(x) is empty or IU(x)1 :5 ,\
(resp. IU(x)1 < w). Each U E U is in one and only one U(x)
and
(#) y < x implies yEU for U E U(x)
Put A = {x EX: U(x) -I 0}. We only need to show I{U(x) :
x E All :5 ,\ (resp. < w). To see this, for each x E A,
take a U(x) E U(x), then there is an open set V(x) such that
x E V(x) C V(x) c U(x) since X is regular. Put W(x) =
V(x) - U{V(y) : y > x}, then {W(x) : x E A} is a disjoint
family of open sets since U is HCP. By (#) and the regularity
of X there is an open set X(x) such that x E X(x) C X(x) c
W(x). The family {X(x) : x E A} is discrete. In fact, for any
x E X, if x E W(y) for some yEA, W(y) only meets X(y). If
xeW(y) for any yEA, then G = X - UyEAX(y) is an open
neighborhood of x since U is Hep and X(y) C W(y) for each
yEA. For any yEA, G n X(y) = 0. Since X satisfies the
D,\CC (resp. DFCC) the discrete family {X(x) : x E A} of
(5) --+(2), (2)
open sets has cardinality :5 ,\ (resp. < w).
--+(3), (3) --+(4), (5) --+(6), (6) --+(7) and (7) --+(4) are obvious.
(4) => (1): Let 9 = {Go: Q E A} be a discrete family of
non-empty open sets of X. If 9 covers X, then it is a locally
finite irreducible open cover of X. Thus 191 :5 ,\ (resp. < w).
Otherwise, take an X o E GOt and an open set VOt such that
X o E VOt C V
o c GOt. Put F = UOtEA VOte then F is a closed set
and U = {GOt: 0 E A} U {X - F} is a locally finite irreducible
DISCRETE CHAIN CONDITIONS AND THE 8-PROPERTY 105
open cover. By (4),191 :5 A (resp.
satisfies the .DACC (resp. DFCC).
< w). This show that
X
Remark 3.3. In [W], it is proved that a regular space satisfies
the DCCC iff every locally finite family of open sets is count­
able. Our Theorem 3.3 generalizes the above result. Noticing
the case DFCC in Theorem 3.3, we obtain
(1) A Tychonoff space satisfies the DFCC iff it is pseudo­
compact (cf. Theorem 3.10.22 of [E].)
(2) A regular. DFCC T1 space is countably compact iff it is
countably paracompact.
4. RELATIONSHIPS
Lemma 4.1. The space X has the B-property (resp. count­
able paracompactness) iff every i1lcreasing open cover (resp.
increasing countable open cover) ojr X has an open star refine­
ment.
Proof: Sufficiency: Let U. = {UOI : 0 < I\,} be an increasing
open cover of X and 9 an open star refinement of U. Put
VOl = U{G E 9 : st(G,9) C UOI } , then V = {VOl: 0 < I\,} is
an increasing open refinement of U satisfying VOl C U01 for any
(} < 1\,. Thus X has the B-property'.
Necessity: Let U = {UOI : 0 < I\,} be an increasing open cover
of X. By Theorem 1 in [V] ( its proof does not use the fact X
is regular and T1 ), U has an open refinement V = {VOl : Q' < I\,}
such that for every 0 < "', VOl C UOI and for every x EX,
there is an open neighborhood Ox of x and an Ox < I\, such
that Ox n (U{VOI : 0 ~ ox}) = 0. Hence st(Ox, V) C U{VOI :
(} < ox} C UOlz • Put WOl = U{O : 0 C X is open and
st(O, V) C UOl } , then W = {WOI : Q < I\,} is an increasing
open cover of X and st(WOI,V) C UOi for any (} < 1\,. Put
9 = {WOl n Vp : 0, {3 < I\,}, then the open cover 9 of X satisfies
for any G E Q, there is a UOI E U such that st{ G, Q) c UOI .
For the countably paracompact case, the proof is similar
noticing that in a countably paracompact space every increas­
106 YINZHU GAO, HANZHANG QU AND SHUTANG WANG
ing countable open cover {Ui : i < w,} has a locally finite open
refinement {Vi : i < w} such that Vi C Ui for any i < w.
Theorem 4.1. If a DACC (resp. DFCC) space X. has the
B-property, then X is A-Lindeliif (resp. compact).
Proof: Suppose not. Take aU = {Ucr : 0 < K} E A,\(X) (resp.
E AF(X)) such that lUI = f~(X) = K (resp. lUI = fF(X) =
K). By Lemma 3.1, cfK = K,. Put Vcr = UP<crUp. Then the
increasing open cover {Va : 0 < K,} has an increasing open
refinement W = {Wo : 0 < K} such that Wo C Vo for any
0' < k since X has the B-property. By Lemma 4.1 W has an
open star refinement Q. Take a non-empty Go E Q and an
00 < K, such that st( Go, Q) c Woo. Since X - Woo t= 0, there is
a G1 E Q and an 01 > 00 such that G 1 = G1 n (X - Woo) # 0
and st( G1 , Q) c W Ot • Thus st( G 1 , Q) c W Ot • Suppose for the
ordinal v, when TJ < V, G'7 and 0'7 have been defined. If 0&/ =
sup{0'7 : TJ < v} < K" then take a Gil E Q and an 011 > all such
that G~ = Gv n (X - WoJ # 0 and st( GVl 0) c Wa ". Thus
st(G 1-£' Q) C WOII. If all -= K" the definition is finished. There
is certainly some T at which the definition is finished. Then
the family {G'7 : TJ < T} of non-empty open sets is discrete.
In fact, for every x EX, if x E X -st(U'7<TG'7' Q), there is a
Gx E Q such that x E Gx and Gx n (U{ G'7 : TJ < T}) = 0. If
x Est(U'7<TG'7' Q), there is G x E Q and TJx < T such that Gx n
G'7% # 0. If TJx # TJ < T, Gx n GTI = 0. To see this, observe that
if v < TJ, st( Gil' Q) Cst( Gil' Q) C W crll and G'7 n WOII = 0. So
st(GII,Q) nG'7 = 0 (and hence st(G'7,Q) nG&/ = 0). Therefore,
st(G II ,Q)nG'7 = 0 for any distinct V,TJ < T. SO if Gx nG'7% t= 0,
then Gx n G'7 = 0 for TJ # TJx· This shows that {G'7 : TJ < T}
is discrete. Since sup{0'7 : TJ < T} = K" {0'7 : TJ < T} is cofinal
in K,. So T ~cfK = K, > A (resp. K 2:: w). This contradicts the
assumption that X satisfies the DACC (resp. DFCC).
Noticing Figure 1 and (*), we have
Corollary 4.1. A T 1 (resp. regular T1 ) space is compact (re­
sp. Lindeliif) iff it has the B-property and satisfies the DFCC
DISCRETE CHAIN CONDITIONS AND THE 8-PROPERTY 107
(resp. DCCC).
Corollary 4.2.
(1) If X is a regular DCCC space, then
B-property ~ Lindelofness ~ paracompactness.
(2) If X is a regular DFCC space, then
B-property ~ compactness +-+ Lindelofness ~ paracompact­
ness.
Corollary 4.3. Let X be a T1 space having the B-property,
then the following are equivalent:
(1) X is wo-Lindelof (resp. compact).
(2) X is WOt+l-compact (resp. countably compact).
(3) X satisfies the DwOtCC (resp. DFCC).
From [D], we know that in a regular space the DFCC (resp.
DCCC) and w-starcompactness (resp. w-star Lindelofness) are
equivalent, so we have
Corollary 4.4. A regularw-starcompact (resp. 'w-star-Lindelof)
space with the 8-property is compact (resp. Lindelof).
Remark 4.1. (1) In Theorem 4.1, hence in its corollaries,
the B-property can not be replaced by the V-property (re­
sp. monotone normality [C), shrinkable property[MN), count­
able paracompactness). Actually, for any Q' ~ 0, [0, w Ot ) has
W = {W(x) : x E X} satisfies chain (F), where W(x) =
{[x,,B] : x ~ ,B < wa}. So by Theorem 3 in [C], [O,wQ) is a
monotonically normal space. It is known that in a monotoni­
cally normal space, every open cover is shrinkable (see Corol­
lary 2.2 in [B]). Since the shrinkable property implies the V­
property, the space [0, w Ot ) has the V-property (hence count­
able paracompactness). In Example 2.1, we have shown that
[O,Wa +2) (resp. [O,Wl)) is WQ+l-compact (resp. w-compact),
and hence DwQCC (resp. DFCC), but it is not wQ-Lindelof
(resp. compact).
(2) From Example 2.1 and Theorem 4.1, we can see that for
any Q' 2:: 0, [0, WQ+l) does not have the 8-property.
108 YINZHU GAO, HANZHANG QU AND SHUTANG WANG
(3) Noticing paracompactness implies the B-property and
the implication is not reversible (see (*)), .Corollary 4.1 im­
proves Theorem 2.3 of [W] (i.e. a regular Tt-space is Lindelof
iff it is paracompact and satisfies the DCCC). Since in a TI ­
space, wI-compact (resp. countably compact) implies the D­
CCC (resp. DFCC) and the implication is not reversible (see
Example 2.2), Corollary 4.1 also improves Theorem 2.1 and
Corollary 2.2 of [Z] (i.e. a regular T1 space is Lindelof (resp.
compact) iff it is WI-compact (resp. countably compact) and
has the B-property).
(4) In [D], the authors gave the following two figures:
compact
Lindelof
1
1
countably compact
WI -
1
compact
1
strongly-1-starcompact
strongly-l-star-Lindelof
1
1
l-s~.arcompact
1 - star-Lindelof
1
1
strongly-2-starcompact
strongly- 2-star-Lindelof
2-starcompact
2 - star-Lindelof
1
.
1
1
1
!
W -
starcompact
Figure 2
!
w -
star-Lindelof
Figure 3
The B-property implies countable paracompactness (see (*)),
which, in turn, implies pseudonormality [Hl. Pseudonormal
(hence countably paracompact ) w-starcompact spaces are known
to be countably compact[Dl, but need not be compact. From
Corollary 4.4, we see that in a regular TI space with the B­
property, all properties in Figure 2 (resp. Figure 3) are equiv­
alent.
DISCRETE CHAIN CONDITIONS AND THE 8-PROPERTY 109
(5) In the Tychonoff space with the B-property pseudo­
compactness (resp, pseudo-Lindelofness[D]) implies compact­
ness (Lindelofness) since a Tychonoff space is pseudocompact
(resp. pseudo-Lindelof) iff it is w-starcompact (resp. w-star­
LindelotiD]) .
Acknowledgement
The authors would like to thank the referee for many helpful
suggestions.
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Changchun Teachers College
China
110 YINZHU GAO, HANZHANG QU AND SHUTANG WANG
Present address
Shimane University
Nishikawatsu-cho, 1060
Matsue, Shimane, Japan
Northwest University
~ian, 710069, China
Northwest University
Xian, 710069, China