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Topology Proceedings Web: Mail: E-mail: ISSN: http://topology.auburn.edu/tp/ Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA [email protected] 0146-4124 c by Topology Proceedings. All rights reserved. COPYRIGHT ° 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. REFERENCES [A] P. S. Alexandroff and P. Uryson, Memoire sur les espaces topologiques compacts, Verh. Akad. Wetensch. Amsterdam 14 (1929), 1-96. [B] Z. Balogh, M. E. Rudin, Monotone normality Topology Appl., 47 (1992), 115-127. [C] P. J. Collins, C. M. Reed, A. W. Roscoe and M. E. Rudin, A lattice of conditions on topological spaces, Proc. Amer. Math. Soc., 94 (1985), 487-496. [D] E. K~' van Douwen, G. M. Reed, A. W. Roscoe and I. J. Tree, Star covering properties, Topology Appl. 39, (1991), 71-103. [E] R. Engelking, General Topology (Heldermann Verlag Berlin, rev.ed., 1989). [H] M. Husek and J. van Mill Recent Progress in General Topology, (North-Holland, 1992). [MN] K. Morita and J. Nagata, Eds., Topics in General Topology ESP.B.V. 1989, 161-202. [MR] J. van Mill and G. M. Reed, Open Problems in Topology, (North Holland, 1990). [T] Y. Tanaka and Y. Yajima, Decompositions for closed maps, Topol ogy Proc., 10 (1985), 399-411. [W] M. R. Wiscamb, The discrete countable chain condition, Proc. Amer. Math. Soc., 23 (1969), 608-612. [Y] Y. Yasui, Some characterizations of a B-property, Tsukuba J. Math., 10 (1986), 243-247. [Z] P. Zenor, A class of countably paracompact spaces, Proc. Amer. Soc., 24 (1970), 258-262. 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