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A note on monotone D-spaces.

1 Introduction

The D-property was introduced by E. K. van Douwen in [5] and was studied widely (for instance, [1], [3], [4], [6] or [7]). A neighborhood assignment for a space X is a function [phi] from X to the topology of X such that x [member of] [phi]{x) for all x [member of] X. A space X is a D-space if for every neighborhood assignment [phi] for X, there is a closed discrete subset F of X such that X = [phi](F) = [intersection]{[phi](x) : x [member of] F}. It is well-known that a space with a point-countable base is a D-space ([1]) and semi-stratifiable spaces are D-spaces ([3], [4]). Hence (T-spaces, stratifiable spaces, Moore spaces and metrizable spaces are all D-spaces.

In [14], the monotone D-property is introduced and studied. A space X is a monotone D-space if for each neighborhood assignment [phi] for X, we can pick a closed discrete subset F([phi]) of X with X = [intersection]{[phi](x) : x [member of] F([phi])} such that if [psi] is also a neighborhood assignment for X and [phi](x) [subset] [psi](x) for each x [member of] X, then F([psi]) [subset] F([phi]). Monotone D-spaces are D-spaces, but the converse is not true. The closed unit interval [0,1] is a D-space, but it is not a monotone D-space ([14]). It is well-known that in generalized ordered spaces the D-property is equivalent to paracompactness ([6]). The Michael line M (the real line with the irrationals isolated and the rationals having their usual neighborhoods), one of the most important elementary examples in general topology, is a paracompact generalized ordered space, and so it is a D-space. In [14], it is shown that the Michael line M is also a monotone D-space.

A linearly ordered topological space is a triple (X, [lambda], [less than or equal to]), where [less than or equal to] is a linear order on the set X and [lambda] is the open interval topology defined by [less than or equal to] (that is, A has a subbase {(a, [right arrow]) : a [member of] X} [union] {([left arrow], a) : a [member of] X}, where (a, [right arrow]) = {x [member of] X : a < x} and ([left arrow], d) = {x [member of] X : x < a}). For a,b [member of] X, (a,b) = {x [member of] X : a < x < b} is called an open interval. The Euclidean space P is a linearly ordered topological space. A generalized ordered space is precisely a subspace of a linearly ordered topological space. It happens that for P = paracompactness (resp., metrizability, Lindelofness and quasi-developability) a generalized ordered space has P if and only if its (minimal) closed linearly ordered extension has P. The main results of the note are as follows.

1. The minimal dense linearly ordered extension of the Michael line is hereditarily paracompact (hence a hereditary D-space), but not a monotone D-space.

2. The minimal closed linearly ordered extension of the Michael line is a monotone D-space.

3. IfX is a D-space (resp., a monotone D-space), so is its Alexandroff duplicate space A(X). Thus A(M) is monotonically Dfor the Michael line M.

Throughout the note, spaces are topological spaces. We reserve the symbols R, Q, P, Z and [Z.sup.+] the set of all real numbers, all rational numbers, all irrational numbers, all integers and all positive integers respectively. Let [phi] and [psi] two neighborhood assignments for a space X, then by [phi] refining [psi] (denoted by [phi] [??] [psi]) we mean [phi](x) [member of] [psi](x) for each x [member of] X. Undefined terminology and symbols will be found in [10].

2 Main results

For the Michael line M, put

l(M) = (R x {0}) [union] (P x {-1,1}).

Obviously the lexicographic order [??] on l(M) is a linear order on l(M). Equip l(M) with the open interval topology generated by the linear order [??] on l(M). Then the Michael line M is homeomorphic to the dense subspace P x {0} of the linearly ordered topological space l(M). The space l(M) is called a dense linearly ordered extension of M. l(M) is also the minimal dense linearly ordered extension of M (see Theorem 2.1 of [13]). Note that the set R x {0} [subset] l(M) with the linearly ordered topology generated by the hereditary order from the order on l(M) is homeomorphic to the Euclidean space R.

It is well-known that the minimal dense linearly ordered extension l(X) of a paracompact space X may not be paracompact, however for the minimal dense linearly ordered extension l(M) of the Michael line M, we have the following Theorem.

Theorem 1. The space l(M) is hereditarily paracompact, and hence a hereditary D-space.

Proof. Let Y be a subspace of l(M). Now we will show that Y is paracompact. Suppose not. Then Y has a closed subspace F homeomorphic to a stationary subset T of some uncountable regular cardinal. Let f : F [right arrow] T be a homeomorphic mapping. Since P x {0} is a discrete open subset of l(M), F \ (P x {0}) is a closed subspace of Y and f(F (P x {0})) is still a stationary subset. So we suppose F [intersection] (P x {0}) = [empty set]. Let [M.sub.1] = l(M) \ (P x {-1,0}) and [M.sub.2] = l(M) \ (P x {0,1}). Put [Y.sub.1] = Y [intersection] [M.sub.1] and [Y.sub.2] = Y [intersection] [M.sub.2]. Let (R, [[tau].sub.1]) be generated by the base [B.sub.1] = [lambda] [union] {[a,b) : a [member of] P, b [member of] P, a < b} and (R, [[tau].sub.2]) be generated by the base [B.sub.2] = [lambda] [union] {(a,b] : b [member of] P, a [member of] R, a < b}, where [lambda] is the usual topology on R. Then for i [member of] {1, 2}, [M.sub.i] as a subspace of l(M) is homeomorphic to (R, [[tau].sub.i]) and thus its subspace [Y.sub.i] can be considered as the subspace of (R, [[tau].sub.i]). Since (R, [lambda]) is second countable it is hereditarily separable. Let [C'.sub.i] be the countable dense subset of [Y.sub.i] considered as a separable subspace of (R, [lambda]) and [C.sub.i] = [C'.sub.i] [union] {y [member of] [Y.sub.i] : y has a predecessor or a successor}. Then for i [member of] {1,2}, the countable [C.sub.i] is dense in [Y.sub.i] and thus [Y.sub.i] as the subspace of (R, [[tau].sub.i]) is separable. Noticing that F = F [intersection] Y = (F [intersection] [Y.sub.1]) [union] (F [intersection] [Y.sub.2]), we see that f(F [intersection] [Y.sub.1]) or f(F [intersection] [Y.sub.2]) is stationary. That is, a closed subset of [Y.sub.1] or [Y.sub.2] is homeomorphic to a stationary subset. Hence [Y.sub.1] or [Y.sub.2] is not paracompact. This contradicts the separability of [Y.sub.1] and [Y.sub.2] (separable generalized ordered spaces are paracompact). In [6] it is shown that in generalized ordered spaces the D-property is equivalent to paracompactness, and thus l(M) is a hereditary D-space.

For the Michael line M, put

[M.sup.*] = (P x {0}) [union] (P x Z).

Let [??] be the lexicographic order on [M.sup.*]. Equip [M.sup.*] with the open interval topology generated by the linear order [??] on [M.sup.*]. Then the Michael line M is homeomorphic to the closed subspace R x {0} of the linearly ordered topological space [M.sup.*]. The space [M.sup.*] is called a closed linearly ordered extension of M ([12]). By Theorem 9 of [16], the space [M.sup.*] is the minimal closed linearly ordered extension of M.

Theorem 2. The space [M.sup.*] is a monotone D-space.

Proof. For a neighborhood assignment [phi]' for [M.sup.*], define a neighborhood assignment [[phi].sup.*] for [M.sup.*] such that [[phi].sup.*] [??] [phi]' as follows. Let [x.sup.*] = <x, k> [member of] [M.sup.*], if x [member of] P, define [[phi].sup.*]([x.sup.*]) = {[x.sup.*]}; if x [member of] Q, then k = 0. Let [I.sub.x] be the maximal open convex subset of [M.sup.*] such that <x,0> [member of] [I.sub.x] [subset] [phi]'([x.sup.*]) = [phi]'(<x, 0>). If [I.sub.x] = [M.sup.*], define [[phi].sup.*] ([x.sup.*]) = [M.sup.*]. Now suppose that [I.sub.x] is one of the following, where [q.sub.x] < x < [r.sub.x], [s.sub.x] < x < [t.sub.x] and {[q.sub.x], [r.sub.x]} [subset] R while {[s.sub.x], [t.sub.x]} [subset] P:

(1) ([left arrow], <[r.sub.x], m>); (2) {<y, i> [member of] [M.sup.*] : y [less than or equal to] [t.sub.x]}; (3) {<y, i> [member of] [M.sup.*]: y < [t.sub.x]};

(4) (<[q.sub.x], j>, [right arrow]); (5) {<y, i> [member of] [M.sup.*] : y [greater than or equal to] [s.sub.x]}; (6) {<y, i> [member of] [M.sup.*] : y > [s.sub.x]};

(7) (<[q.sub.x], k>, <[r.sub.x], l>); (8) {<y, i> [member of] [M.sup.*]: [s.sub.x] [less than or equal to] y [less than or equal to] [t.sub.x]};

(9) {<y, i> [member of] [M.sup.*] : [s.sub.x] < y < [t.sub.x]};

(10) {<y, i> [member of] [M.sup.*] : [s.sub.x] < y [less than or equal to] [t.sub.x]} [union] {<[s.sub.x], i> : i [greater than or equal to] k};

(11) {<y, i> [member of] [M.sup.*] : [s.sub.x] [less than or equal to] y < [t.sub.x]} [union] {<[t.sub.x], i> : i [less than or equal to] l};

(12) {<y, i> [member of] [M.sup.*] : [s.sub.x] < y < [t.sub.x]} [union] {<[t.sub.x], i> : i [less than or equal to] l};

(13) {<y, i> [member of] [M.sup.*] : [s.sub.x] < y < [t.sub.x]} [union] {<[s.sub.x], i> : i [greater than or equal to] k}.

Then define

[[phi].sup.*]([x.sup.*]) = {<y, i> [member of] [M.sup.*] : y < [r.sub.x]} if (1) holds;

[[phi].sup.*]([x.sup.*]) = {<y, i> [member of] [M.sup.*] : y < [t.sub.x]} if one of (2) and (3) holds;

[[phi].sup.*]([x.sup.*]) = {<y, i> [member of] [M.sup.*] : [q.sub.x] < y} if (4) holds;

[[phi].sup.*]([x.sup.*]) = {<y, i> [member of] [M.sup.*] : [s.sub.x] < y} if one of (5) and (6) holds;

[[phi].sup.*]([x.sup.*]) = {<y, i> [member of] [M.sup.*] : [q.sub.x] < y < [r.sub.x]} if (7) holds;

[[phi].sup.*]([x.sup.*]) = {<y, i> [member of] [M.sup.*] : [s.sub.x] < y < [t.sub.x]} if one of (8) to (13) holds.

For x [member of] r, put [phi](<x, 0>) = [[phi].sup.*](<x, 0>) [intersection] (R x {0}). Then [phi] is a neighborhood assignment for the subspace R x {0} of [M.sup.*]. Since M is monotonically D and is homeomorphic to the subspace R x {0} of [M.sup.*], there is a closed discrete subset [F.sub.[phi]] of M such that R x {0} = [phi]{[F.sub.[phi]] x {0}) and if [psi] is a neighborhood assignment for R x {0} with cp [??] [psi] then [F.sub.[phi]] [contains] [F.sub.[psi]].

Put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if x [member of] P, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; if x [member of] Q, then there are [a.sub.x], [b.sub.x] [member of] Q such that x [member of] ([a.sub.x], [b.sub.x]) and ([a.sub.x], [b.sub.x]) [intersection] [F.sub.[phi]] = [empty set] since [F.sub.[phi]], is closed in M. So [x.sup.*] [member of] W = (<[a.sub.x], 0>, <[b.sub.x], 0>) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is closed in [M.sup.*]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If x [member of] Q, then k = 0 and there are [c.sub.x], [d.sub.x] [member of] Q such that x [member of] ([c.sub.x], [d.sub.x]) and ([c.sub.x], [d.sub.x]) [intersection] [F.sub.[phi]] = {x}. Put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; if x [member of] P, put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is discrete in [M.sup.*].

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [phi]([F.sub.[phi]] x {0}) = R x {0}, there is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that <y, 0> [member of] [phi](<x, 0>) = [[phi].sup.*](<x, 0>) [intersection] (R x {0}). Assume x [member of] P. By the definition of [[phi].sup.*], [[phi].sup.*](<x, 0>) = {<x, 0>} and hence x = y, contradicting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So x [member of] Q. By the definition of [[phi].sup.*](<x, 0>), [phi](<x, 0>) is one of the sets ([left arrow], [r.sub.x]) x {0}, {[q.sub.x], [right arrow]) x {0}, ([s.sub.x], [t.sub.x]) x {0} and R x {0}, where x < [r.sub.x], [q.sub.x] < x and [s.sub.x] < x < [t.sub.x]. Hence [y.sup.*] = <y, k> [member of] [[phi].sup.*](<x, 0>). So [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [union] <{[phi]' ([y.sup.*]> : [y.sup.*] [member of] [F.sup.*.sub.[phi]']} = [M.sup.*] since [[phi].sup.*] [??] [phi]'. If [psi]' is a neighborhood assignment for [M.sup.*] with [phi] [??] [psi], then obviously [F.sup.*.sub.[phi]'], [subset] [F.sup.*.sub.[psi]']. Thus [M.sup.*] is monotonically D.

Theorem 3. The space l(M) is not a monotone D-space.

Proof. Assume that l(M) is monotonically D. Define a mapping f : l(M) [right arrow] R, where R is the Euclidean space, as follows: for each <x, i> [member of] l(M), f(<x, i>) = x. Then f is continuous. In fact, for an open interval (a, b) of P and <x, i> [member of] [f.sup.-1]{{a,b)), since x [member of] (a,b), there are [q.sub.x], [r.sub.x] [member of] Q such that x [member of] ([q.sub.x] ,[r.sub.x]) [subset] (a, b). Thus <x, i> [member of] (<[q.sub.x], 0>, <[r.sub.x], 0>) [subset] [f.sup.-1]((a, b)). So [f.sup.-1](a,b)) is open in l(M). To show that f is closed, let F' be a closed subset of l(M) and x [not member of] f(F'). If x [member of] Q, then [f.sup.-1](x) = {(x, 0)} and <x, 0> [not member of] F'. So there is an open interval G = (<[c.sub.x], 0>, <[d.sub.x], 0>) of l(M) with <x, 0> [member of] G and G [intersection] F' = [empty set], where [c.sub.x], [d.sub.x] [member of] Q. Thus x [member of] ([c.sub.x], [d.sub.x]) and ([c.sub.x], [d.sub.x]) [intersection] f(F') = [empty set]. If x [member of] P, then [f.sup.-1](x) = {<x, -1>, <x, 0>, <x, 1>} and [f.sup.-1](x) [intersection] F' = [empty set]. Since F' is closed in l(M), we can take [a.sub.x], [b.sub.x] [member of] Q such that (x, -1) [member of] (<[a.sub.x] ,0>, <x, 0>) with (<[a.sub.x] ,0>, <x, 0>) n F' = [empty set] and <x, 1> [member of] (<x, 0>, ([b.sub.x], 0>) with (<x, 0>, <[b.sub.x], 0>) [intersection] F' = [empty set]. Thus x [member of] and ([a.sub.x], [b.sub.x]) [intersection] f(F') = [empty set]. Hence f(F') is closed. Since l(M) is monotonically D, its closed continuous image P is monotonically D (see Theorem 1.7 of [14]). Because that the monotone D-property is closed hereditary, the subspace [0,1] of P is monotonically D. However Theorem 2.3 of [14] shows that [0,1] is not monotonically D. A contradiction. []

Recall that a space X is meta-Lindelof if every open cover of X has a point-countable open refinement.

Example 4. There is a monotone D-space which is not a meta-Lindelof space.

Proof. Let N = [Z.sup.+] and N = {Ns C N : [absolute value of [N.sub.s] = [omega], s [member of] S}, where S [intersection] N = [empty set], be infinite such that [N.sub.s] [intersection] [N.sub.s], is finite if s [not equal to] s' and that N is maximal with respect to the last property, that is, is the maximal almost disjoint family of N. Define a topology [tau] on X = N [union] S by the neighborhood system {B{x) : x [member of] X}, where B(x) = {{x}} if x [member of] N and B(x) = {{s} [union] ([N.sub.s] \ F) : F [subset] N, [absolute value of F] < [omega];} if x = s [member of] S. Put [PSI](N) = (X, [tau]). Since the set of all isolated points of [PSI](N) is N and the subspace S of [PSI](N) is discrete, [PSI](N) is a monotone D-space ([14]) (so a D-space). However [PSI](N) is not meta-Lindelof ([2]). []

Let X be a space, A [subset] X and U be a family of subsets of X, put sf(A, U) = [st.sup.1]{A, U) = U{U [member of] U : U [intersection] A [not equal to] [empty set]}. Inductively [st.sup.n+1](A, U) = [union]{U [member of] U : U [intersection] [st.sup.n] (A, U) [not equal to] [empty set]}. A space X is [omega]-star Lindelof ([8]) if for every open cover U of X, there is n [member of] [Z.sup.+] and a countable B [subset] X such that [st.sup.n] {B, U) = X.

Theorem 5. The Michael line M is not an co-star Lindelof space.

Proof. Let Q = {[q.sub.1], [q.sub.2], [q.sub.i],...} and for each [q.sub.i] [member of] Q, the open interval [I.sub.i] containing [q.sub.i] be with the length less than 1/[2.sup.t]. Then U = {[I.sub.i] : i [member of] [Z.sup.+]} [union] {{p} : p [member of] P} is an open cover of M. For any countable subset B of R, T = R \ ([union]{[I.sub.i] : i [member of] [Z.sup.+]} [union] B) is uncountable. Take [t.sub.0] [member of] T, then for any n [member of] [Z.sup.+], [t.sub.0] [not member of] [st.sup.n]{B, U). So M is not an [omega]-star Lindelof space.

A space X is [[omega].sub.1]-compact if every closed discrete subset has cardinality < [[omega].sub.1].

Remark 6. (1) An [[omega].sub.1]-compact D-space X is Lindelof for the D-space X, l(X) = e(X) ([9]). By [[omega].sub.1]-compactness of X, e(X) = [omega] and thus /(X) = [omega].

(2) A space is Lindelof if and only if it is [[omega].sub.1]-compact and meta-Lindelof note that every point-countable open cover of the [[omega].sub.1]-compact space has a countable subcover (Lemma 7.5 of [11]).

(3) The Michael line M cannot be the following: strongly n-star-Lindelof, n-star-Lindelof, [[omega].sub.1]-compact or Lindelof by Theorem 5 and Fig. 4 of [8].

Since M is a meta-Lindelof D-space without Lindeofness, the [[omega].sub.1]-compactness condition in (1) and (2) cannot be removed.

The Alexandroff duplicate space A(X) for the space X is the set X x {0,1} equipped with the topology as follows: points in X x {1} are isolated and each point (x, 0) in X x {0} has the basic neighborhoods as the form: (U x {0,1}) \ {(x, 1)}, where U is an open neighborhood of x in X. The following Lemma is obvious.

Lemma 7. Let X be a space. Then if F is a closed set in X, F x {0,1} is closed in A(X); if D is a discrete set in X, D x {0,1} is discrete in A(X).

Theorem 8. Let X be a space. Then X is a D-space if and only if A(X) is a D-space; X is monotonically D if and only if A(X) is monotonically D.

Proof. Sufficiency: let [psi] be a neighborhood assignment for A(X). If X is a D-space, for each (x, 0) [member of] A(X), take an open [U.sub.x] in X containing x with ([U.sub.x] x {0,1} \ {(x, 1)}) C [psi]((x, 0)). Then for the neighborhood assignment {[U.sub.x] : x [member of] X} for X there is a closed discrete subset F of X such that X = [union]{[U.sub.x] : x [member of] F}. By Lemma 7 F' = F x {0,1} is a closed discrete subset of A(X) and A(X) = [union]{[psi](z) : z [member of] F'}. So A(X) is a D-space. If X is monotonically D, for each x [member of] X, put

[B.sub.x] = {x} [union] {y [member of] X : y [not equal to] x and {<y,0>, <y,1>} [subset] [psi](x,0>)}.

Take an open [U.sub.x] [subset] X containing x with ([U.sub.x x {0,1} \ {(x, 1)}) [subset] [psi]((x, 0)), then [U.sub.x] [subset] [V.sub.x] and thus x [member of] [V.sup.o.sub.x]. Put [psi]x(x) = [V.sup.o.sub.x]. Then the neighborhood assignment [psi]x for X satisfying that ([psi]x(x) x {0, 1}) \ {<x, 1>} [subset] [psi](<x, 0>). So there is a closed discrete subset [F.sub.[psi]x] of X such that X = [union]{[psi]x(x) : x [member of] [F.sub.[psi]x]}. For the closed discrete subset [F.sub.[psi]] = [F.sub.[psi]x] x {0,1} of A(X), it holds that A(X) = [union]{[psi](z) : z [member of] [F.sub.[psi]]}. The rest proof of the sufficiency is obvious.

Necessity: note that the D-property and the monotone D-property are closed hereditary and the closed subspace X x {0} is homeomorphic to X. []

In the following corollary, M, R, S, P, C and [0, [[omega].sub.1]] are the Michael line, the Euclidean space, the Sorgenfrey line (the real line with the half-open intervals of the form [a, b) as a basis for the topology), the Niemytzki plane, the Cantor set and the usual ordinal space respectively.

Corollary 9. A(M) is a monotone D-space; A(R), A(S), A(P), A(C) and A([0, [[omega].sub.1]]) are D-spaces, but not monotone D-spaces.

Proof. M is monotonically D ([14]). Clearly R, S, P, C and [0, [[omega].sub.1]] are D-spaces. By [14], S, C, [0, [[omega].sub.1]] and [0, 1] are not monotonically D. Since R has a closed subspace [0, 1] and P has a closed subspace [0,1] x {1} homeomorphic to [0,1], R and P are not monotonically D. Hence by Theorem 8, the conclusion of the corollary is true. []

Remark 10. (1) For the Michael line M, A(M) has a point-countable base: put [B.sub.q] = {((a,b) x {0,1}) \ {<q, 1>} : a, b [member of] Q, a < q < b}, q [member of] Q. Then B = [union]{[B.sub.q] : q [member of] Q} [union] {{<x, 1>} : x [member of] R} [union] {{<p, 0>} : p [member of] P} is a point-countable base for A{M).

In general, a Moore space may not be monotonically D. For a first countable [T.sub.2]-space X, let x [member of] X and {[B.sub.n]{x) : n < [omega]} be fixed basis of x with [B.sub.n+1]{x) [subset] [B.sub.n](x), n < [omega]. Define a topology v on M(X) = X [union] (X x [omega]): points of X x [omega] are isolated; a basic neighborhood of x [member of] X is the form [C.sub.m]{x) = {x) [union] {<y, n> : (n [greater than or equal to] m) [conjunctioin] (y [member of] [B.sub.n](x))}, m < [omega]. Then (M(X), v) is a Moore space ([15]).

(2) The Moore space (M(X), v) is monotonically D: since the subspace X of all non-isolated points of (M(X), v) is discrete, by Theorem 1.7 of [14] (M(X), v) is monotonically D.

Acknowledgment. The authors are very grateful to the referee for many helpful comments and suggestions, especially for outlining the proof of Theorem 1 which improves the result in the previous version of the paper.

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Yin-Zhu Gao ([dagger])

Wei-Xue Shi

* The project is supported by NSFC (No.10971092).

([dagger]) Corresponding author

Received by the editors November 2008--In revised form in March 2010.

Communicated by E. Colebunders.

2000 Mathematics Subject Classification : 54D20,54F05,54C25,54E18.

Department of Mathematics

Nanjing University

Nanjing 210093

China

email: yzgao@nju.edu.cn, wxshi@nju.edu.cn
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Author:Gao, Yin-Zhu; Shi, Wei-Xue
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:May 1, 2011
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