aWctkmwise normal and countably paracompact if every open ad refinement. A sufficient condition for paracompsctnerfs, il;~

finite covef”s, is given, and is agplied to the problem of the

WT!J): F&nary 54L 20,54018,54D15; Seconda’ry 54B10,54E~5.

Tamano [IO, 111, Katuta [3] and ions of garacompactness in terms ng linearly ordered covkrs Katuta

&it ‘wnWons an a pzkracompact space X X witi any paracompact space’be paracom-

cor;tiriue the studies of gatuta [S], Vaughan [ 151

al qaces are assumed to be 7’1. Let X be a topological covering {R, : 4 E A} having the

rid ta Bc: 8 pr&se refinement uf $9 if R, c G, for each a in A. itim set A haa been linearly ordered by <. Ilhen, following

if, for each a in A, the coillection

thb foElowing 9~0 theorems:

tudied certain types of parawmpact @&&of t&&Aogi- e defined a subset A of a space X to be a&-~&iom~ct

any colbection of open subsets of X whose union contains (in X) collection 9@ of open bets *hose unionz contains

, there exists a G in 9@ for which R e 6. 1

n ar -paracompact subset, we now phrase Telgarsky’s concept f a space X as follows: a subset A of ~a space X,ic said to

lrovided that whenever Y is any paracompact space, A x Y t of the product X x Y.

well-situated so ‘that the following theorem of 113)

t X be a regular space which has an open, order locally as a precise refinement W by clrosed well-situated subsets

kal product of X with arty paracom,pact space is pamcompact.

lI$ space Xu-focaliy compact provided that X is the union ed locally compact subspaces, and we call a space Y R Es any non-empty closed subspace of Y, then R

tm that y has a compact neighbourhood in R, Eveq locally ttered and, from results in [9] and El33 one easily

mpact M-space, then X is C-scattered iff X is a-locally eorem C lies in Tel&sky’s theorem in [l3] &at a

8 paracompact space X is well-situate4 in X modest generalization of order local finiteness, and remain true for point order ioca;Uy finite collections,

n [ 151. The coU ctztion %!I is linearly &c&y at for each a in A, the a~llection

rovided there exists a function

regdar space

Considering the nature of Theorems A ami 1

is a regular space such. that every open refinement, then X is paracompact. Vaughar,\ disp that tie space 96 of all countab1.e ordinak with the uau property that every open cover of J2 has an opt:n order cu We note that the space 9k is collectionwke normal and count not m-par4compact whenever m is greater than 0. In if X is a regular space such that every open cover of X refinement, the: r X must be coltectionwise nor

L,et kf denc:e the Michael line, that is, A4 is irrational points made discrete. Let *< be any well-orderin of 1M Such that if x is a rational number and y is an irrat,ion Then the collection of all singletons, as ordered1 b of M by compact sets. Since M is paracompact b space of irrationals ia not normal [S], we co generalized by removing the open order locally finite co Morita established the following.

eorwa E (Morita). Let X be a paracompact M-space.

X with any paracdtmpact space is puracompact (equioalent, X has 4 v-locally finite csvaring by compact sets).

Since every a-locally Ifinite collection may be linearly ord finite, and in particular $will be order locally &&space having an order lc cahy finite cove o-locally finite covering ?q compact sets. 1 least the case for metric spaces.

C:Sven afL open cover $P of X >rr: S, there exists an open aWet =(H,: 8EB)

that for each b in El, them is a natural nmbcr n(b) 8 that the mm

h& :qj, .v,bx(2-‘) , . , s , Mb x{(n(b)- 1)-l} P .kG !& x{O, )s (bj-‘, (n(b)+ 1)-l, . . .I

rc each t :).‘ptaGne.J in some member of $.

Thir paragraph, nqt a main part of the proof, d&bes a simple general con&ruc- Gne which wilt be referred to in ithe fofiowirig paramaph. If.@ = {et: i B (4 6)) is any

red coi%etiion such that to etlch set Gj is amorated a finite number of sets, say G,, we may form an ordered collection ,w by replacing each set Gi

. . . , Gi,p; here we take i, n ~j, m if i e j in (I, 6) or, in case

iI in S, let lil, = fl&]. men fi9] = {Ha : a e A) c X We have associated

ith each set R, in 9R 8 finite number of sets, namely R, x(l), R, x(2”‘}, . . . t R, x

(C&a)-- l)-‘) and R, X10, n(a)?, (#i!(a)+ 1)” , . . .}; (here the. integer n (a) is equal to n(b) where b E B satisfies ~~I&] = 4ia, j. Now form t$e ordered col&tion 98’, ti in the preceding paragraph, by replacirtg each R, E 4ie by the finite number of sets BP x(1.), R, x(2-7,. . . which we just enumerated,

Define a map g : BY-) 89 as bolIows: Let R, x F be an arbitrary memberof a’; here F is one of the sets {I), {L, ‘) , . . . , ((n(a)- 1)“) or {0, n(&, . . .}. Define g(R. x F’) = Ea where Ea is any metn,ber of 8 which contains Ha x F = f(&] x R

tt is straightforward to show that 3’ is an order cushioned refinement of 8 via the map g, completing the proof.

We next, establish a sufficient condition for a ‘space to be colIec%onwise normal,

2.2. If X is a regular space slrch that m?ry open covering of X has ati qm, mdc? cushioned refinement, then X is mllectiomvise normal.

Lzt .X be a regular space such that every open cover of X has an Open order iined refinement. Let {F,: y E r) be a discrete collection of dowd sub& of A: neF=L_@..: y~r).ForeachyhlrletK,~E:CFv.ForeachxQF;let V’be

n neighbourhood of x wch that cl( VW> n K,, = 8). Let

s is an open covering of X so there exists an open cmhioned refbment 9’

.S C)} be the subsystem 1.

Ckxmier lb clcmd s& F, If R, &4! -and &.n&, # 8, then define

Let G(&) P~{G~: C?a II Fv #a}. Fm sod Fe are distinct sets in {F$ y E T}. We shatl show th

0 I&c R, and Rb be members of i%i mch that R, n Fe it 0 and Rb n F@

R, + Rb We may suppose &at Q < b without loss of general&y. Since R, some V, in _S an4 Vz n.Fa PC fd, then ct( V’i) nJ$” B so that c!(Ra) n F” since Q <b, by definition of Gb we have R, n @, = 8. Since CL Q R G, n Gb =8. It follows that G(F,) ana: G(F@) are disjoint.

We shall now show that Fv c G(F,.) for each y in E Let x E F,. Then ther some R, in 9 such&at x E.R~. Suppose

Since $V is order cushioned in gq necessarily x ~f[Rb] = V,, for some b Rb n Fv = 8, Since &, n F, =r 8, thtree exists P, Z .& with Rb n & # 0; th

that Vv n F, = 0. This is a contradiction since x E Vv and x E F,,, sa that n x E Ga. It follows that Fv C= G&J, completing the proof.

In [Z], Dowker showed that if the product of a Hausdorff space R with unit intervs! is normal, then R is normal and countabfy paraornp proof shows that we may use the space S of Lemma 2.1 in place of the ct interval. This fact, together with Lemmas 2.1 and 2.2 yield the followin

In [4], Krajewski showed that a space is paracompact iff it is &r&n expandable. This, together with Theotern 2.3, yields the followin

C~rouiary 2*4. A regular space X is paracompact iff X is &refinable and covtv of X hns an open, srdet cushioned refinemerrt.

The ideta! of order tocal finiteness may be slightly ~e~~raliz~d

some member of (G=: a: e &I}, then there exist8 an &UWnt b in hood V of cs” sueh that x E 413~ and V meets o

Let 3 = {G*: a E {A, <)} be a point ord Given an a in #I., let H,! denote the set neighbourhood which merrts at most finitely Ha may be empty, but, ea<:h set H, is open and finite and covers X. Theorem A may now

Theorem A’ (Matuta). A regdar space X is pamcompuct iff e has an open, point order kdly finite refinement.

Let X be a regular spaa having a point order locally finite that 3 has a precise re5neruent ST by compact sets. If we now order locally finite open re:%rement %’ as in the p&r it is not clear that x is a refinement of Sl! nor is it cle a precise refinement of % by c.omp:ict sets. Nevertheless, we product theorem is true for point order locally finite covers. The key b sufficient condition for paracompactness.

Theorem 3.2. 1 X is a ~i-egular space which has uoo o_pe~ cotiering 3 such that $9 has a precise refinement by CIY paracompact.

Proof. Let 99 = (C&: a E (A, a); be an open point order 1 re;gular space X and 9 = {4)Q: a E A} be a co a-paracompact and Pa c G, for every o in A. ofX.ForeachainA,let~~={HnG,:H~ an a-paracompact subset of X, there exists an that Pa c Ua, and such that each member of ZO. Let 92 = Ll{a,: a E A). Linearly order eat linearly order 9 as follows if D, E E !R, say D E Eifd<~inA;ifd=e,thenletDprecedeE linear ordering which hBs been assigned to

It 3s straightforward to show that R, as ju finite refinement of SK By Theorem A’, the space X is proof.

The following version of Theorem. C follows

3.3 (Telgarsky), I ct X be Q mpdtw

H. W. Mmh / lLi~arr!3 odedcavenr 313

t (A, c)} be un open; poitit order iocally finite revering of } be 8 precke refinement of, {G&: It GrA$ :by &sod well-situated su

Y=(G, x Y: u EA) isr an open, point order locally finite cover x Y: Q E A} is a precise refinenrat tif 9 by cy garacomgact subsets

rxomp44ctness of X x Y now follows ~?rom TheoremN3.2, complet-

t&red subsets of paracompact spaces are well-situ&ted, 2.5 of [13] is also true. for point order local finiteness:

y). Let X be a pamcompaet space having an ape-t, point ord that 4$1 has a precise re&ememt by closed C-scattered

n the tvpological product of X with any pamcompact space is also

We end this sectkn with two observations. The first, Exsmple 3.5, shows that not true if we replace the closed C-scattered subspaces by closed

pact subspaces, and the second, mieorem 3.6, shows that if we replace -scattered subspaces in Theorem 3.4 by closed locally compact sub-

&en Theorem 3.4 essentially reduces to Katuta’z Theorem B.

&!k (The Mchael line). Let M denote the Michael line, Q denote th ;ti numbers and P denote the set of irrational numbers. Let P be wells

be the well-ordered open covering of M whose first member iz y the singleton irrationals. The colktion 3 is order Pocally finite.

is a countable closed subset of M, Q is closed and a-locally compact, so as a precise refinement t y closed, cr-locally compact subspaces, but ‘the

with the space of in ational real numbers is not normal 181.

Let X be a paracompac~ space which has un open, point order Eoc~ally d beally fictite) coveritg 9 such that +? has a precise refinemen by closed,

an open, poitit ordflr ioca/ly finite (order lmcl fly

se refinemeat by qxmpact sets.

ji be an open point order vering of X, and 9’ = (Pa : a E A) be a precise

Ily cornpa& Sinlce .X is .P, is the sum of a locally rxe X is paracompact

open 1ocrfIy finite family = {Hb(d): b e &) stxch in &. Let 8 = {.!‘&(a): b E B, and a E A}. L~~~~~,r~y

As noted in the introduction, there exists a paracumpact slpaoe X h locally finite covering by compact seti% such that tie pro&@ .of X paracompact space rnay fail $0 be normal. Theorem 42 Mow 40 Theorem E shows tbat if a metric space X has a point ordl

compact sets, then the pr&tict of X with my parae&npact ct. it is intereskg to contrast this result with the M!owin~ therm of

[ 141: if X is paracompact and has a fl-closure preserving CuVer by cornpaa sets, then the product of X with any paracompact spa- must be paracompact,

We first establish the folkxw6ng sirmple lemma.

Proof. Let d be a compatilble metric for X and let Z = {KY: a E (A, S)) be an O&T Iocstlly finite collection of compact subsets of X For each natural cumber i, let &(i) =(x: d(x, K,,)< i-‘1. L&Q ={&: if b <a, thenK&n& =@}L Sincefa&et Ka is compact and % is order locally finite and pairwise disjoint, it is &ar that for each a in A, there is an integer i such that KG e Su;. It is also clear that each ooikction Z, is discrete, so that x is ar-discrete, completing the proof.

Using Lemma 4.1, ~;ore prove the following.

Theiwem 4.2. Let X be a mtrizabk spklce. If X has 42 p&2 t ordct locally finite cmi?ri?ig by compact sets, then X has Q cr-d&we cawing by compact se&

Prod. Let X=(Ka: a E (A, s)) be a point or&z loMy Rnita covering,af X bby compact sets. For each a in A, define G, to be the set of all poifits x iti K , such that x has a neighbourhood which meets only finitely many members of the collection & : b < a}. Set 3 = ( Ga : a E (A, +). Note that C#J is an order 1ocMy fiti~ cowr’of

X arzd that each set G, is open relative to the subspace Ka. Sin{* C& fs ~relatively ,we have GQ=U(G,(n): n =a,re,...}~whereieachset E,(n)is ed in

C&(n) is compact, Let %” = {(3&): o E (A., ai)jw Tkm b an ally finite collection of compact sets. tn I= C&(n) -bj{G~fn): b S a). Since 2!$ isorderlocally

subspace C&(m) so that J&(n) = u{.&(n, 5): i = I, 2,. * .I

,i if? order locally tin@. $3 th@t -

Q] CS. Ati& para#H8npact SubSet3, hi% SecCmd Prague Tapokqiod Sympwiufi, { [2] (F.-W. l%wbk~~ Ck oa\intatilly pWcompact spwes, Cabad. S. ;Watlr. 1 (1951) 21 l3-1 X Katu& A tMmwp 0’. w%u!&pactness of prodt%t spaces, Prw . Japan Acad, 4 [4] L. Ictajcwbkr, On wcpawng XocaIly finite ccrflections, Canad. J. Math. 23 (197 [s] &A, ‘kichae!, A nate on rwzzqrnpact spaces, PFOC, Amer. Math. &TX. 4 (1954) 831-83 [61 ?.A. lb!%&wf, Anotbei n&e cm parawmpaet spaces, PFQC. Amet. Msttb. SW. 8 (1957) [7] EA. h&ha& Yet arwtber note oki ?aracompa.zt spaces, Prw. AYW. %fatx Sot. 10 ( 1959~ f8f E.A. htfichd* The product bf a normal space and a metric space need not be n

Mat& Sot. 69 (196,2) 375-376. [9] K. Writ&, Qn the grodtrct of par&compact spaces, Proc. Japa11 Acad. 39 (1963) 559-563.

[lOI HI Tatnanrl, Note on pcompact~~q J. Math. Kyoto Univ. 3 (1963) 137-143. [l 1) H. T.&no, 0n : une characterizati Bns of paracompactness, Topology Conference, Ari

thivetity, ‘Rmpe, AT, 1967,277.285. [12] )i. Tamano and J.E. Vaughan, Parawmpactness and elastic spaces, Broc. Amer. Math. kc:. 2s

(1971) 299-303. [13] R. Te@kalcy, C-scattered and paracompact spaces, Fund. Math. 73 (1971) 59-74. 1141 R. Tel@sky, Spaces &f&d by topological games, Fund. Math. 88 (1975) 193-223. [IS] J.E. Vau&an, Linearly ordered collections and paracompactness, Proc. Amer. Math. Sac. 2

186192.