DISCRETE SEQUENCES OF POINTS

Transcription

1 Volume 3, 1978 Pges DISCRETE SEQUENCES OF POINTS by J. E. Vughn Topology Proceedings Web: Mil: Topology Proceedings Deprtment of Mthemtics & Sttistics Auburn University, Albm 36849, USA E-mil: ISSN: COPYRIGHT c by Topology Proceedings. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume DISCRETE SEQUENCES OF POINTS J.E. Vughn Abstrct We consider wek version of R. L. Moore's property D. Roughly speking, spce X is sid to hve property D if ech discrete (in the loclly finite sense) sequence of points in X cn be "expnded" to discrete fmily of open sets in X. A spce is sid to hve property wd if ech discrete sequence hs subsequence which cn be "expnded" to discrete fmily of open sets. All regulr, submetrizble spces nd ll relcompct spces hve property wd. In the clss of regulr spces in which every point is Go' propert wd is both hereditry nd wi-fold productive. In the clss of T 3 -spces, finite to-one perfect mps preserve property D, but do not necessrily preserve property wd (for exmple, we show tht certin finite-to-one perfect imges of the Niemytzski plne nd of the Pixley-Roy spce do not hve property wd). Property wd, however, is preserved by n-to-one perfect mps for every positive integer n. Whether every product of perfectly norml T1-spces hs property wd is independent of the usul xioms of set theory.

3 238 Vughn 1. Introduction R. L. Moore introduced property 0 in his book [M0, 2 p. 69] nd this concept hs been rediscovered nd renmed severl times since then [GJ, Problem 3L], [H], [Mis], [Mol]. As fr s we know, property wd ws first explicitly considered (under different terminology) by K. Morit [Mol]' nd lter in different context by others [VD ], [V ]. Both 1 3 properties del with countbly infinite sets of points which re discrete (in the whole spce) in the loclly finite sense. Such sets re often clled countble, closed discrete sets, nd we sometimes refer to them s discrete sequences (for more precise definitions see 2). Definition 1. A countbly infinite discrete set A c X hs ppopepty D in X provided there exists discrete fmily of open sets {U : E A} such tht U n A {} for ll E A, nd the discrete set A is sid to hve ppopepty wd in X provided there exists n infinite subset of A which hs property D in X. Definition 2. (resp. ppopepty wd) A spce X is sid to hve ppopepty D provided tht every countbly infinite discrete set in X hs property D (resp. property wd) in X. Property 0 cn be considered s wek form of normlity from t lest two points of view. Severl uthors hve noticed tht T 3 l/2-spce X hs property D if nd only if

4 TOPOLOGY PROCEEDINGS Volume every discrete sequence in X is C-embedded in X, nd tht T 3 -spce X hs property D if nd only if every pir of disjoint closed sets, one of which is discrete sequence, cn be seprted by disjoint open sets. Anlogous chrcteriztions of property wd cn be found by strting with either of the two preceding chrcteriztions of property D, nd "pssing to subsequence." Most wys of looking t property wd re equivlent in the clss of T 1/2-spces, but differ 3 in generl. For exmple the definition of property wd given by R. E. Hodel [Ho] is slightly stronger thn the one in this pper becuse Hodel's definition pplies to "sequences hving no cluster points" insted of to "discrete sequences" (these two definitions re obviously the sme in the clss of T l spces). Since fter pssing to subsequence, property wd does the sme thing s property D, it retins some of the strength of property D. Indeed, some results which hve property D s prt of their hypotheses re still true when property D is replced by property wd. This is the cse with Theorems 159 nd 161 in Moore's book [M0, p. 70] nd Theorems 11 nd 2 14 in [H]. In ddition to the study of C(X), property wd hs been used concerning mpping theorems [SA], crdinl functions in topology [Ho], set theory nd topology [VDl]' nd countb1y compct extensions of X in SeX) [Moll, [K 2 ]. The following digrm of implictions (which holds in the clss of T -spces) indictes the reltion of property 3 wd to severl well-known properties.

5 240 Vughn / norml" prcompct pseudonorml~propertyd~property wd countbly "" countbly / compct ~ prcompct We believe tht property wd is of specil interest in spces in which every point is Go. If we do not hve the requirement tht every point is Go' then property wd is neither hereditry nor finitely productive in the clss of T 3 1/2-spces. For exmple, the Tychonoff plnk [Wi, p. 122] is esily seen not to hve property wd, nd yet it is subspce of compct T -spce (clerly, every (countbly) com 2 pct spce hs property wd). Thus, property wd is not hereditry in the clss of T 3 1/2-spces. We cn lso use known exmples to show tht property wd is not productive in tht clss either. It is known [GS, Exmple 5.2] tht there exist two countbly compct spces X nd Y (both subspces of Sew)) with the property tht X x Y is pseudocompct (i.e., there exists no infinite loclly finite fmily of open subsets of X x Y) but X x Y is not countbly compct (so X x Y contins discrete sequence). Thus, both X nd Y hve property wd but X x Y does not. Our principl theorems re concerned with the preservtion of property wd under perfect mps, nd with finding clsses of spces in which property wd is hereditry or is productive to some extent. We originlly proved severl of our theorems in the clss of T 3 -spces in which every point is Go' but discovered tht we did not require the full strength of tht clss. For these results, we only need

6 TOPOLOGY PROCEEDINGS Volume the following concept (which is known under severl nmes): A point x in spce X is clled Husdorff-Go provided tht {x} is the intersection of countbly mny closed neighborhoods of x (we ssume the open neighborhoods re nested). In 3 we prove Theorem A. In the clss of spces in which every point is Husdorff-Go' property wd is wi-fold productive. In contrst to this we note tht even in the clss of first countble, T 3 1/2-spces, property D is not productive. For exmple, the Sorgenfrey line S [Wi, 4.6] is first countble, T 3 1/2-spce which hs property D (S is Lindelof spce) but S x S does not hve property D (Sorgenfrey's proof [So] tht S x S is not norml uses two disjoint closed discrete sets, one of which is countble). The spce S x S hs property wd (by Theorem A, or by Corollry 3.5 which sttes tht every submetrizble T 3 -spce hs property wd); so S x S is n exmple of spce which hs property wd but does not hve property D. For n esy exmple of Moore spce which does not even hve property wd we my tke the well-known exmple N u ~ of S. Mrowk [Mr ] (this spce is clled ~ in [GJ]). Recll tht this l spce consists of countble discrete spce N nd mximl fmily ~ of lmost disjoint subsets of N. The points of N re isolted nd locl bse for point R in ~ consists of ll sets of the form {R} U T where T is cofinite subset of R. The spce N U ~ is pseudocompct but not countbly compct; so it does not hve property wo. In 4. we prove

7 242 Vughn Theorem B. In the clss of Urysohn spces in which every point is Husdorff-Go' property wo is hereditry. Agin note the contrst in behvior of property wo with tht of property 0: Property 0 is not hereditry in the clss of first countble, T 3 1/2-spces. To s'ee this, let I* denote the top nd bottom lines of the lexicogrphiclly ordered squre (cf. [VI]). We my consider the Sorgenfrey line S s subspce of I * nd therefore S x S s subspce * * * of I x I. Since I is first countble, compct T 2 -spce, * * * * so is I x I. Thus I x I hs property 0, but its subspce s x S does not. In 5, we give severl results concerning perfect mps. We mention one theorem nd one exmple here. Theorem C. Let f: X ~ Y be perfect mp from spce X which hs property wo onto T 3 -spce Y. If there exists positive integer n such tht for ll y E Y we hve If-l(y) I < n, then Y hs property wo. The restriction in Theorem C on the crdinlity of point inverses is needed. Exmple 5.6 shows tht there exists finite-to-one perfect mp defined on the Niemytzki plne (which is submetrizble T 3 1/2-spce) whose imge does not hve property wo. (This imge spce is n exmple of seprble Moore spce which does not hve property wo). In 6, we show tht Exmple 5.6 nd similr exmples (e.g., the Pixley-Roy spce [PRJ) cn be used with property wo in the study of perfect imges of submetrizble nd relcompct spces.

8 TOPOLOGY PROCEEDINGS Volume Some of the results in this pper were nnounced in 2. Preliminries We first recll severl definitions. 2.1 Definitions. A fmily J of subsets of topologicl spce X is sid to be loclly finite t point x in ~ provided there exists n open set U contining x which intersects t most finite number of elements of J. If there exists n open neighborhood of x which intersects t most one member of J, we sy tht J is discpete t x. A subset A ex is clled discpete set if the fmily {{}: E A} is discrete t every point of X. tn this pper we re concerned with the cse where A is countble, nd we sometimes sy tht A is discpete sequence. In other words, discrete sequence of points is one-to-one indexing of countbly infinite discrete set. We note tht spce in which every point is Go' is Tl-spce, nd tht in Tl-spce every subset of discrete set of points is closed set. We mke the following definition in order tht we my consider in detil stndrd type of construction. 2.2 Definition. A point p in spce X is clled Husdopff point (resp. Upysohn point) provided tht for every x E X\{p} there exist open sets U, V such tht x E U, P E V, nd U n V = ~ (resp. IT n V = ~). Clerly, point which is Husdorff-Go is Husdorff point (this is why we use the term "Husdorff-Go"). Recll

9 244 Vughn tht spce is sid to be Urysohn spce if every pir of distinct points hve disjoint closed neighborhoods Lemm. Let A be countbly infinite subset of spce x, nd let p E A\A. If P is both Husdorff-Go nd Urysohn point in x, then there exists n infinite sequene { i : i < w} in A nd fmily V = {Vi: i < w} of open sets in X such tht. ] point of X\{p}. E V. 1 iff i = j, nd V is discrete t ech Proof Let {p} = n{g.: i < w} where G. is n open set 1 1 contining p for ech i < w. Pick O E GO n A, nd let V o nd U be open sets with O EV, p E U O ' nd V n U = ~. o o o There exists positive integer no such tht O i G Pick no l E (U O n G ) n A (so ~ O ll nd let VI nd U l be open no sets such tht l E VI C U n G ' p E U l C o U O ' nd no VI n U l =~. Thus V n VI =~. Continue in this wy to construct the sequence { l : i < w} nd the fmily of open sets V = {Vi: i < w} whose closures re mutully disjoint nd such tht. E V. c G (where no < n l < < n. < 1 1 n i - l 1-1 n. < ). The condition V. c G implies tht V is 1 n 1 i-i loclly finite, hence discrete, in X\{p} Remrk. In Lemm 2.3, if we ssume tht p is Husdorff-Go but not necessrily Urysohn point, then we cn proceed in mnner similr to the proof of Lemm 2.3, nd get the fmily V = {V.: i < w} to be loclly finite 1 fmily of mutully disjoint open sets, but since the closures of the members of V re not necessrily disjoint, it does not follow tht V is discrete fmily. This sitution occurs in Bing's countble, connected Husdorff spce [B].

10 TOPOLOGY PROCEEDINGS Volume Likewise, if we only ssume in Lemm 2.3 tht the point p is Go nd Urysohn point, then we cn get the elements of V to hve mutully disjoint closures, but we cnnot necessrily get V to be loclly finite t every point in X\{p}. This sitution occurs in spce considered by R. M. Stephenson, Jr. [St., Exmple 5] which is Urysohn spce in which every point is Go but one point is not Husdorff-Go 2.5. Lemm. A regulr Tl-spce X hs property wd if nd only if X stisfies the following condition: For every countbly infinite discrete set A c X, there exists n infinite loclly finite fmily {Vi: i < w} of open sets in X such tht Vi n A ~ ~ for infinitely mny i < w. Proof. If X hs property wd, it stisfies the condition without ny ssumption concerning seprtion xioms. We ssume the condition holds nd tht X is regulr Tl-spce, nd show tht X hs property wd. By pssing to subsequence of A, nd by using the fct tht every subset of A is closed in the Tl-spce X, we my ssume tht Vi n A consists of single point, cll it i' for ll i < w. Let no = O. There exists positive integer n < wnd n open neighborhood l U of f) such tht U n V. Proceeding by no no 1 induction, we get loclly finite fmily {Un. n V : i < w} 1 ni of mutully disjoint open sets such tht n. E U n. n V n for ll i < w. Since X is regulr there exist open sets Wi such tht ni E Wi C Wi C U n V Then {Wi: i < w} ni ni is discrete fmily of open sets in X such tht W. n A = { } for ll i < w. Thus X hs property wd. This 1 n. 1 completes the proof.

11 246 Vughn Recll tht spce is clled feebly compct provided tht every loclly finite fmily of open sets in the spce is finite, nd tht feeble compctness is equivlent to pseudocompctness in T 3 1/2-spces Remrk. A Tl-spce is countbly compct if nd only if it is feebly compct nd hs property wo. This cn be considered s slight generliztion of [GJ, 3L, 5]. A spce X with topology T is clled submetrizble provided there exists metrizble topology M on X such tht M ct. We recll well-known exmple of non-regulr, submetrizble spce. Let M denote the usul topology on the closed unit intervl [0,1] nd let T be the topology on [0,1] hving s subbse M u {[O,l], {lin: 1 ~ n < w}} The topology T is obviously submetrizble nd not regulr. This is n exmple of submetrizble spce which does not hve property wo. One wy to see this is to note tht this spce is feebly compct but not countbly compct. 3. Some Bsic Results 3.1. Proposition. If X is first countble, Husdorff spce hving property wd, then X is regulr. Proof. set F nd If X is not regulr, then there exist closed point p i F such tht for every neighborhood U of p, IT n F ~ S. Let {u : n < w} be nested locl bse n t p such tht {p} = n{it : n < w}. Clerly, for infinitely n mny n < w, we hve (Un "IT 1) n F ~ S, nd by the nesting, n+

12 TOPOLOGY PROCEEDINGS Volume these subsets of F re mutully disjoint. By pssing to subsequence, we my ssume tht (Un \ U + ) n F ~ ~ n l for ll n < w. distinct points. Pick sequence n E (Un 'U + l ) n F of n Now A = {n: n < w} is discrete sequence in X, nd hence by property wd (nd by pssing to subsequence) we my ssume there exists discrete fmily 1/ = {V : n < w} of open sets in X such tht V n A {n } n n for ll n < w. Since {U : n < w} is still locl bse for p, n there exists k < w such tht Uk intersects t most one member of V. On the other hnd, for ll n < w, Un n V ~ ~; n so Un n V ~~. Thus, for n ~ k, Uk n V ~~. n n This is contrdiction. The hypothesis "Husdorff" cnnot be wekened to "T " l in Proposition 3.1, s the exmple of the cofinite topology on w shows. Proposition 3.1 cn be considered s generliztion of C. Aull's result [AI] tht first countble, countbly prcompct T 2 -spce is regulr. The following exmple, which is n elbortion of Aull's exmple [A Exmple 1], 2, shows tht "first countble" cnnot be in ExmpZe. A submetrizble spce which hs property wd nd is not regulr. The set X consists of wi x (w + 1) together with point p not in wi x (w + 1). All points of wi x wre to be isolted, nd point (,w) where < wi hs bsic neighbor hoods of the form {(,m): n ~ m ~ w}

13 248 Vughn Let {P : n < w} be prtition of wi n into countbly mny uncountble sets. Define the bsic neighborhoods of p s follows: For every < wi nd n < W W(,n) = {(B,i): :S B < wi' n.:s i < w, BE UP.} U {pl. j~n J Since ech W(,n) contins points whose second coordinte is w, the spce X is not regulr. To see tht X is submetrizble, note tht the sets W(,n) re clopen in X, nd tht {W(D,n): n < w} serves s bse for p in metrizble topology On X in which ll the other points hve their sme neighborhoods. Tht the spce X hs property wd (even property D) follows from the fct tht for every co~ntble set A c X with PiA, there exists < wi such tht A n W(,D) =~. It is obvious tht property wd is closed-hereditry, i.e., inherited by closed subspces. We now consider the hereditry nture of property wd Proposition. Property wd is hereditry in the clss of Urysohn spces in which every point is Husdorff- Proof. Let Y c X, nd let A be countbly infinite discrete set in the subspce Y. If A is lso discrete in X f there is nothing to prove; so we ssume tht there is point P E X\Y such tht p E A\A. By Lemm 2.3, there is sequence of distinct points { : i < w} c A nd fmily V = i {Vi: i < w} of open sets with Vi n A = {il for ll i < w, nd such tht V is discrete t ech point of X\{p}. Since P i Y, the fmily {V. completes the proof. 1 n Y: i < w} is discrete in Y, nd this It follows from Proposition 3.3 tht property wd is

14 TOPOLOGY PROCEEDINGS Volume hereditry in the clss of regulr spces in which every point is Go. By Proposition 3.1, therefore, property wd is hereditry in the clss of first countble, Husdorff spces. We hve n exmple which shows tht property wd is not hereditry in the clss of Urysohn spces in which every point is Go. This leves us with the following Problem. Is property wd hereditry in the clss of spces in which every point is Husdorff-Go? The exmple ([0,1], T) given in 2 is non-regulr, submetrizble spce, which does not hve property wd. The next result shows, however, tht every regulr, submetrizble spce hs property wd Proposition. Let (X,S) be spce hving property wd in which every point is Husdorff-Go. If T is regulr topology on X which is finer thn S (i.e., T ~ S) then (X,T) hs property wd. Proof. Let A ex be countbly infinite discrete set in the spce (X,T). If A is lso discrete in the spce (X,S) there is nothing to prove; so we ssume tht A is not discrete set in (X,S). By pssing to subsequence (if necessry), we my ssume tht there exists point P E C1S(A)\A. By Remrk 2.4, there exists n infinite set { i : i < w} e A nd fmily {Wi: i < w} c S which is loclly finite in (X,S) t every point of X\{p}, nd such tht. E W. iff j = i. Since (X,T) is regulr, there exists J 1 U E T such tht p E U nd CIT(U) n A =~. Put Vi (Wi\C1T(U)) for ll i < W. Then {Vi: i < w} is loclly finite fmily of open sets in (X,T) such tht Vi n A ~ S for

15 250 Vughn ll i < w. Lemm 2.5 implies tht (X,T) hs property wd, nd this completes the proof Corollry. Every regul~ submetrizble spce hs property wd Corollry. Let X be first countble, submetrizble spce. Then X hs property wd if nd only if X is regulr. Exmple 3.2 shows tht "first countble" cnnot be deleted from Corollry Products The theory of products of spces hving property wd is closely relted to the theory of products of countbly compct nd feebly compct spces Lemm. Let {X : < K} be fmily of Tl-spces hving property wd, let A be n infinite countble discrete subset of X = TI{X : < K} nd put Y = Cl ~ x (A», where TI is the usul projection mp TI : X ~ X () If there exists < K such tht Y is not feebzy compct, then A hs property wd in X. (b) C,Jnversely, if A hs property wd in X, nd ech point in ech X is Husdorff-Go' then there exists < K such tht Y is not feebly compct. Proof of (). If Y is not feebly compct, there exists n infinite loclly finite fmily {U : i < w} of open sets in i Y Since TI{A) is dense in Y, we my pick by induction n infinite sequence Xi. E U i. n TI(A). Since X is T I, the ] ]

16 TOPOLOGY PROCEEDINGS Volume loclly finite set {x. : j < w} of points is discrete 1. ] sequence in Y nd in X. By pplying property wd nd pssing to subsequence, we my ssume tht there exists discrete fmily {Vi: i < w} of open sets in X such tht x. E V. for 1. J J ll j < w. It is esy to check tht {1T -1 (V j ) : j < w} is discrete in X nd tht ech 1T- l (V'.) n A ~ ~. Since every J subset of A is closed in X, we cn find n open subset of ech 1T- l (v.) tht contins exctly one point of A. J Proof of (b). Suppose A hs property wd in X. Then no subspce of X cn both contin A nd be feebly compct. If ech Y is feebly compct, then by Remrk 2.6 ech Y is countbly compct (since property wd is hereditry to closed subspces). Further, since ech point in Y is Husdorff Go' Y is first countble; hence sequentilly compct. By Theorem 5.2 in [SS] every product of sequentilly compct spces is feebly compct. Thus IT{Y : < k} is feebly compct nd contins A. This is contrdiction Theorem. If {X : < wi} is fmily of spes hving property wd, nd in whih every point is Husdorff- Go' then TI{X : < wi} hs property wd. Proof. Let A be n infinite countble discrete set in X = IT{X : < wi}. By Lemm 4.1 (), it suffices to show tht there exists some Y which is not feebly compct. If is feebly compct, nd pro this is flse, then every Y ceeding s before, we see tht every Y compct. is sequentilly By theorem of Scrborough nd Stone [SSe Theorem 5.5], every wi-fold product of sequentilly compct spces is countbly compct; so A is contined in countbly

17 252 Vughn compct subspce of X. This contrdicts the ssumption tht A is discrete in X. The next result does not require tht points be Go-sets Theopem. Let P be topologicl ppopepty which is hepeditpy to closed subspces, nd such tht evepy countbly compct spce hving ppopepty P is compct. Then evepy ppoduct of Tl-spces, ll of which hve both ppopepty P nd ppopepty wd, hs ppopepty wd. Ppoof. The proof is similr to tht of Theorem 4.2, but uses the Tychonoff product theorem [Wi' 17.8] rther thn the theorem of Scrborough nd Stone. We stte few consequences of this result Copollpy. () (b) Evepy ppoduct of metpic spces hs ppopepty wd. Evepy pelcompct spce hs ppopepty wd. wd, (c) Evepy ppoduct of pegulp, -spces hving ppopepty hs ppopepty wo. (d) (MA +,CH). Every product of perfect, T -spces 3 hving ppopepty wd hs ppopepty wd. (e) (MA +,CH). Evepy ppoduct of pepfectly nopml T -spces hs wd. 2 Ppoof. For (d) use the theorem of W. Weiss [W] which sttes: (MA +,CH) implies tht every perfect, countbly compct T 3 -spce is compct. Then (e) is consequence of (d) Thus in the clss of spces in which every point is Husdorff-Go' there re mny fmilies whose products hve

18 TOPOLOGY PROCEEDINGS Volume property wd. Finding fmily in tht clss whose product does not hve property wd is equivlent to solving we11 known open problem in generl topology Lemm. The following re equivlent. wd 3 (1) There exists fmily {x : < K} of spces hving nd in which every point is Husdorff-Go3 such tht X = IT{X : < K} does not hve property wd. (2) There exists fmily {Y : < K} of first countble3 sequentilly compct T 2 -spces whose product Y IT{Y : < K} is not countbly compct. Proof (1) ~ (2). If X does not hve property wd, there exists n infinite closed discrete set A c X such tht A does not hve property wd in X. By Lemm 4.1() ech Y = C1 (TI(A)) is feebly compct nd therefore sequentilly x compct nd first countble. Since A c Y = IT{Y : < k} we see tht Y is not countb1y compct. (2) ~ (1). By Theorem 5.2 of [55] the product spce Y is feebly compct but by hypothesis not countb1y compct. Thus Y does not hve wd by Remrk 2.6. Using this lemm, we cn point to product spces which do not hve property wd. Under certin set theoretic ssumptions such s CH, severl uthors hve given exmples to show tht the product of first countble, sequentilly compct, T 3 -spces need not be countb1y compct. As we noted in 4.5, such products do not hve property wd becuse they re feebly compct. Using the set theoretic ssumption 0 nd the technique of A. Ostszewski [0], we constructed [V 2 ] fmily of first c~untble, perfectly norml, sequentilly compct

19 254 Vughn T 2 -spces whose product is not countbly compct. By combining this exmple with Corollry 4.4(d) we hve 4.6. Proposition. The sttement "every produt of perfetly norml, T 2 -spes hs property wd" is independent of the usul xioms of set theory (ZFC). The theory of box products of spces hving property wd hs been considered by Eric vn Douwen, who lso proved independently some of the results in this section [VD ] Perfect mps In this section we show tht in the clss of T -spces, 3 property wd is not necessrily preserved by finite-to-one perfect mps unless there is n upper bound on the crdinlity of the point inverses. On the other hnd, property is preserved by ll finite-to-one perfect mps. 5:1. Proposition. Let f: X -+- Y be perfet mp from spe X hving property wd onto regulr Tl-spe Y. If there exists positive nturl number N such tht If-l(y) I ~ N for ll y E Y, then Y hs property wd. Proof. Let T be countbly infinite discrete set in Y. By Lemm 2.5, it suffices to show tht there exists n infinite loclly finite fmily V of open sets in Y such tht V n T ~ ~ for infinitely mny V in V. To prove this, we will use the following well-known fct bout perfect mps: If 5 is loclly finite collection in X, then {f(s): S E 5} is loclly finite in Y. Since f is closed mp it suffices to find loclly finite fmily {U t : t E T} of open sets in X such tht f-l(t) cut for infinitely mny t E T.

20 TOPOLOGY PROCEEDINGS Volume By pssing to subsequence, we my ssume tht there exists positive integer m ~ N such tht If-l(t) I m for ll t E T. The proof now proceeds by induction on m. We will do the cse m = 2, nd leve the reminder of the proof (including the cse m = 1) to the reder. Let f-l(t) = {t,b } for ll t E T. Since {f-l(t): t E T} is discrete t fmily in X, the sets A = {t: t E T} nd B = {b : t E T} t re discrete ets in X. By property wd there exist n infinite set T I c T nd discrete fmily {V : t E T I } of open t sets in X such tht V n A = {t} for ll t E T I Next we t pply property wd to {b : t E T I }, nd get n infinite sub t set T I Ie T I nd discrete fmily {W : t E Til} of open t sets in X such tht W t n B = {btl for ll t E T I I. Then {V U W : t E Til} is loclly finite fmily of open sets t t in X such tht f-l(t) c (V U W ) for ll t E Til. This t t completes the proof Proposition. If f: X + Y is perfect mp such tht f-l(y) is finite for ll y E Y, X is Tl-spce hving property D, nd either X or Y is regulr, then Y hs property D. Proof. Since regulrity is preserved by perfect mps, it suffices to prove the result for the cse in which Y is regulr. If T is countble, discrete set in Y, then since X is Tl-spce, S = U{f-l(t): t E T} is countble, discrete set in X. By property D, there exists discrete fmily {V : s E S} of open sets in X such tht V n S = is} s s -1 for ll s E S. Put U = U{V : s E f (t)}. Then t s {Inty(f(U»: t E T} is loclly finite fmily of mutully t

21 256 Vughn disjoint open sets in Y such tht Inty(f(U )) n T = {t} for t ll t E T. Since Y is regulr, this completes the proof. Next we show tht properties D nd wd re reflected by perfect mps Proposition. If f: X ~ Y is (qusi) perfect mp from Tl-spce X onto spce Y hving property wd, then X hs property wd. Proof. Let A be countbly infinite discrete set in X. Since f is closed mp, f(a) is discrete set in Y. Since f-l(y) is (countbly) compct for ll y E Y, we see tht f-l(y) n A is finite for ll y E Y. In prticulr, f(a) is infinite. Since f is continuous, every discrete -1 fmily of open sets in Y cn be brought bck by f to discrete fmily of open sets in X. Thus, there exists discrete fmily V of open sets in X such tht V n A is nonempty nd finite for every V E V. Since every subset of A is closed in X, we my refine V to get discrete fmily of open sets in X, ech of which contins exctly one point of A. In similr mnner, one cn show tht property D is reflected by (qusi) perfect mps f: X ~ Y where X is Urysohn spce. The reminder of this section is concerned with showing tht finite-to-one perfect mps cn destroy property wd. In prticulr, we will show tht certin finite-to-one perfect imges of the Niemytzki plne, nd of the Pixley-Roy spce, do not hve property wd Lemm. Let R be closed discrete subset of

22 TOPOLOGY PROCEEDINGS Volume spce X, nd {F : < k} fmily of mutully disjoint sub sets of R. Let p: X ~ Y be the quotient mp which collpses ech F to single point nd is one-to-one on X\U{F : < k}. Then p is closed mp. Further, p is perfect if nd only if ech F is finite. (The proof is routine.) 5.5. Lemm. Let R be closed discrete subset of Urysohn spce X, nd B countble fmily of infinite subsets of R hving the property (*) for every countble HeR, if H n B ~ ~ for ll B E B, then H does not hve property D in X. Then there exists perfect mp p: X ~ Y such tht Y does not hve property wd. Proof. Let B = {B i : i < w} nd let {F : i < w} be i fmily of mutully disjoint subsets of R such tht (1) F i is finite for ll i < w, nd (2) if j ~ i then F. n B. ~ ~. 1 J Let p: X ~ Y be the quotient mp which collpses ech F to i point. By Lemm 5.4, p is perfect mp, nd clerly {p(f ): i < w} is countble discrete set of points in Y. i We show tht this set does not hve property wd in Y. it did hve property wd in Y then there would exist subsequence {p(f. ): j < w} nd open sets V. in Y such tht 1 j ] () p(f.) E V. for ll j < w, nd (b) {V.: j < w} is disi J J J crete in Y. Now (b) implies tht {p-l(v.): j < w} is dis J crete in X, nd () implies tht F. is finite subset of J p (V ). Since X is Urysohn spce, we my put the points j of F. 1 j into open sets which re contined in V. nd hve ] disjoint closures. This shows tht F = U{F. : j < w} hs 1 j property D in X. By (2), F intersects ech B E B, nd thus If

23 258 Vughn F does not hve property D in X. This contrdiction completes the proof. pplied. We now give severl exmples of how Lemm 5.5 my be 5.6. Exmple. Let N be the Niekytzki plne. Recll tht N is the set of ll points in the upper hlf plne: N = {(x,y): y ~ OJ. Let R denote the x-xis: R = {(x,o): - 00 < x < oo}. The topology on N is defined s follows: Points bove the x-xis hve their usul open disks s bsic neighborhoods, nd points p on the x-xis hve s bsic neighborhoods ll sets of the form {p} U A (where A is n open disk in N tngent to the x-xis t p). Let B be countble bse for the usul topology on R. We show tht (*) of Lemm 2.2 holds for Rnd B. Suppose tht H is countble subset of R such tht H n B ~ ~ for ll B E B. To show tht H does not hve property D in N, it suffices to show tht Hnd R\H cnnot be seprted by open sets in N. This is done in the stndrd wy using Bire ctegory rgument. In similr mnner, Lemm 5.5 dn be pplied to S x S, where S is the Sorgenfrey line, nd to mny versions of Niernytzki's spce such s the next exmple Exmple. (R. W. Heth [He]) Let H be Heth's V-spce. Recll tht the set H = {(~,y): y ~ } nd the topology for H is defined s follows. All points (x,y) with y > re isolted, nd for point p = (PO'O) on the x-xis, locl bse is given by ll sets of the form

24 TOPOLOGY PROCEEDINGS Volume V(p,n) {p} U {(x,y): 0 < y < lin, nd y = ± «1T14) x - (1T14) PO) }. where 0 < n < w. Thus, V(p,n) is subset of the union of two lines through Po hving slopes 1T14 nd -1T14. Clerly, the sme choice of Rnd B s in 6.6 shows tht (*) of Lemm 5.5 holds for the spce H Remrk. Heth's V-spce H is homeomorphic to closed subset of the Pixley-Roy spce (this lso hs been noticed by D. Lutzer nd H. Bennett). Recll tht the Pixley- Roy spce A consists of the set of ll finite subsets of the rel line R with the following topology. For ech x in A nd ech open set U in the usul topology on R, put [x,u] {y E A: x eye U}. The collection of ll such [x,u] forms bse for the topology on the Pixley-Roy spce [PR]. We show tht H is homeomorphic to Z = {x E A:lxl ~ 2 nd x ~ S}. Clerly, Z is closed in A. We show tht Z is homeomorphic to the copy of H obtined by rotting H by 45 in the plne. Let H' = {(x,y): x ~ y} with ll points (x,y) with x < y isolted, nd for points (x,x) tke s locl bse ll sets of the form V«x,x),n) = {(x,y): x ~ y < lin} U {(y,x): x ~ y < lin} where 0 < n < w. Clerly H' is homeomorphic to H. Define mp f: H' + Z by f«x,x» = {x} nd f«x,y» = {x,y} if x < y. Since f is one-to-one nd f (V ( (x, x), n»= [{ x}, (x-lin, x + lin)], it follows tht f is the desired homeomorphism Exmple. The Pixley-Roy spce stisfies (*) of

25 260 Vughn Lemm 5.5. This follows t once from Remrk 5.8 becuse A hs closed subspce which stisfies (*) of 5.5. This shows tht A does not stisfy property D, nd this slightly generlizes the well-known results tht A is not pseudonorml, therefore neither norml nor countbly prcompct (cf.[vd ]). 2 Further, this shows tht there exists spce Y which is perfect finite-to-one imge of A, which does not hve.property wd. The spce A is hereditrily metcompct Moore spce (cf. [VD ]), nd both these properties re preserved by per 2 fect mps [Wo ], [W0 ]. Thus Y is he~editrily metcompct l 2 Moore spce (hving the countble chin condition) which does not hve property wd. 6. Some New Uses of Property wd In this section we show how Exmple 5.6 nd similr exmples cn be used concerning how perfect mps destroy submetrizbility nd relcompctness. Recll tht T 3 1/2-spce is clled relcompct if it is homeomorphic to closed subset of product of copies of the rel line. It is esy to see tht every product of rel lines hs property wd, nd hence every relcompct spce hs property wd (Corollry 4.4b). The fct tht relcompct spces hve property wd hs been noted erlier in [SA] nd [VD ], nd it lso follows from bsic fcts bout 3 relcompctness nd Morit's chrcteriztion of property wd given in [Mol]. Recll gin tht spce X with topology T is clled 8ubmetrizble provided tht there exists metrizble topology M on X such tht MeT. As we hve mentioned, every regulr, submetrizble spce hs property

26 TOPOLOGY PROCEEDINGS Volume wd (Corollry 3.5). It is known tht submetrizble T 1/2 3 spces of crdinlity c hve the following three properties: (i) they re relcompct [GJ, 8.17 nd 15.24], (ii) they hve Go-digonl (cf. [BL]), nd (iii) they hve property wd (for two resons). We now show tht perfect mps cn destroy submetrizbility by destroying nyone of these three properties A perfect mp defined on submetrizble spce which destroys the Go-digonl, but preserves relcompctness nd property wd. F. G. Slughter, Jr. [S] nd V. Popov [P] hve shown tht there exists perfect (2-to-l) mp defined on the disjoint union of two copies of the Michel line whose imge does not hve Go-digonl. Since the imge is prcompct spce of crdinlity c, it is relcompct [K] nd hs property wd A perfect mp defined on submetrizble spce which destroys relcompctness, but preserves the Go-digonl nd property wd. S. Mrowk [Mr ], [Mr ] hs constructed l 2 (2-to-l) perfect mp, defined on the disjoint union of two copies of the Niemytzki plne, whose imge is not relcompct. This imge does hve Go-digonl since both the domin nd rnge re Moore spces [Wo 2 ], nd the imge hs property wd by Proposition A perfect mp defined on submetrizble spce which destroys property wd (nd therefore relcompctness) but preserves the Go-digonl. Exmple 5.6 in this pper shows tht there exists (finite-to-one) perfect mp whose domin is the Niemytzki plne nd whose imge does not hve

27 262 Vughn property wd. The imge hs Go-digonl, nd of course is not relcompct. In order to nswer severl questions of S. Mrowk, Akio Kto [K ] constructed exmples of first countble l spces which show tht 6.4. A spce which is the union of countble, closed discrete set nd relcompct set need not be relcompct, nd 6.5. There is finite-to-one perfect mp f: X + Y which destroys relcompctness nd such tht I{y E Y: If- l (y) I ~ 2} I = w. The exmples constructed in 5, cn be used to show tht both 6.4 nd 6.5 hold (Eric vn Douwen hs informed me tht R. Pol hs lso constructed simple exmple to show tht 6.4 obtins, but Pol's exmple is not first countble). Let f: N + Y be the finite-to-one perfect mp constructed on the Niemytzki plne N s in Lemm 6.5. The mp f destroys relcompctness becuse it destroys property wd. Further, there re only countbly mny y E Y such tht If -1 (y) I ~ 2. This shows tht 6.5 obtins. To see tht 6.4 obtins use the imge spce Y. Let A = {y E Y:lf-l(y) I ~ 2} nd B Y\A. Then A is countble, closed discrete subspce of Y (Lemm 5.4) nd B is relcompct becuse it is homeomorphic to subspce of Niemytzki'sspce N. Thus Y = A U B, but Y is not relcompct. To get nlogous exmples for 6.4 nd 6.5 concerning N-compctness insted of relcompctne~s, we use (insted of

28 TOPOLOGY PROCEEDINGS Volume the Niemytzki plne) Heth's V-spce (see Exmple 6.7) or Mrowk's N-compct version of Niemytzki's spce [Mr ]. 2 Ileferences [B] [BL] [GJ] [GS] [H] [He] [HO] [K] C. E. Au11, A note on countbly prcompct spces nd metriztion, Proc. Amer. Mth. Soc. 16 (1965), , A certin clss of topologicl spces, Prce Mt. 11 (1967), R. H. Bing, A countble connected Husdorff spce, Proc. Amer. Mth. Soc. 4 (1953), 474. D. K. Burke nd D. J. Lutzer, Recent dvnces in the theory of generlized metric spces, Topology Pro ceedings, Memphis Stte University Conf. Mrcel Dekker, New York, E. K. vn Douwen, Functions from the integers to the integers nd topology, (to pper)., The Pixley-Roy topology on spces of subsets, Set-theoretic Topology, G. M. Reed, Ed., Acdemic Press, New York, 1977, pp , Personl communiction. J. Dugundji, Topology, Allyn nd Bcon, Inc., Boston, L. Gillmn nd M. Jerison, Rings of continuous functions, vn Nostrnd, Princeton, J. Ginsburg nd V. Sks, Some pplictions of ultrfilters in topology, Pcific J. Mth. 57 (1975), J. D. Hnsrd, Function spce topologies, Pcific J. Mth. 35 (1970), R. W. Heth, Screenbility, pointwise prcompctness nd metriztion of Moore spces, Cndin J. Mth. 16 (1964), R. E. Hodel, The number of closed subsets of topologicl spce, Cndin J. Mth. 30 (1978), M. Ktetov, Mesures in fully norml spces, Fund. Mth. 38 (1951),

29 264 Vughn [K l ] [K 2 ] [Mi] [Mis] [Mol] A. Kto, Union of relcompct spces nd Lindelof spces, (preprint)., Vrious countbly-compctifictions nd their pplictions, Generl Topology nd Appl. 8 (1978), E. Michel, Topologies on spces of subsets, Trns. Amer. Mth. Soc. 71 (1951), V. V. Miskin, Closed mps of Moore spces, Notices Amer. Mth. Soc., vol. 24, no. 6 (1977), Abstrct 77T-GllO, A-558. K. Morit, Countbly-compctifible spces, Sci. Reps. Tokyo Kyoiku Digku, sec. A, vol. 12, no. 314 (1972), [M0 ] R. L. Moore, Foundtions of point set theory, Amer. 2 Mth. Soc. Colloquium Publictions, vol. XIII, revised [Mr l ] [Mr 2 ] edition, Providence, S. Mrowk, On completely regulr spces, Fund. Mth. 41 (1954), , On the unions of Q-spces, Bull. Acd. Polon. Sci., Mth. Astr. Phys., vol. 6, no. 6 (1958), [Mr ], Some comments on the uthor's exmple of 3 non-r-compct spce, Bull. Acd. Polon. Sci., Mth. Astr. Phys., vol. 18, no. 8 (1970), [0] A. Ostszewski, On countbly compct perfectly norml [PRJ [P] [SS] spces, J. London Mth. Soc., (2) 14 (1976), C. Pixley nd P. Roy, Uncompletble Moore spces, Proc Auburn University Conf., Auburn, Al., V. Popov, A perfect mp need not preserve Go-digonl, Generl Topology nd Appl. 7 (1977), C. T. Scrborough nd A. H. Stone, Products of nerly compct spces, Trns. Amer. Mth. Soc. 124 (1966), [SA] M. K. Singl nd S. P. Ary, On theorem of Michel [S] Morit-Hni, Generl Topology nd its Reltions to Modern Anlysis nd Algebr IV, Soc. Czech. Mth. nd Phys., Prgue, 1977, F. G. Slughter, Jr., A note on perfect imges of

30 TOPOLOGY PROCEEDINGS Volume [So] spces hving Go-digonl, Notices Amer. Mth. Soc. 19 (1972), A-807. R. H. Sorgenfrey, On the topologicl product of prcompct spces, Bull. Amer. Mth. Soc. 53 (1947), [St] R. M. Stephenson, Jr., Symmetrizble 3 J- 3 nd wekly [VI] first countble spces, Cn. J. Mth. 29 (1977), J. E. Vughn, Lexicogrphic products nd perfectly norml spces, Amer. Mth. Monthly, 78 (1971), [V 2 ], Products of perfectly norml 3 sequentilly compct spces, J. London Mth. Soc. (2) 14 (1976), [V ], A wek version of property V, Notices Amer. 3 Mth. Soc. 24 (1977), Abstrct 77T-G120, A-560. [W] W. A. R. Weiss, Countbly compct spces nd Mrtin's [Wi] Axiom, Cndin J. Mth. 30 (1978), S. Willrd, Generl Topology, Addison Wesley, Reding, [Wo 1 ] J. M. Worrell, The closed continuous imge of metcompct topologicl spces, Port. Mth. 125 (1966), [W0 2 ], Upper semi-continuous decompositions of developble spces, Proc. Amer. Mth. Soc. 16 (1965), University of North Crolin t Greensboro Greensboro, N.C