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On Countably Metacompact Spaces
Hayashi, Yoshiaki
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Bulletin of University of Osaka Prefecture. Series A, Engineering and nat
ural sciences. 1960, 8(2), p.161-164
1960-03-30
http://hdl.handle.net/10466/7939
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http://repository.osakafu-u.ac.jp/dspace/
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161'
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On Countably Metacompact Spaces
Yoshiaki HAyAsHi*
(Received January 29, 1960)
A space X is called countably metacompact if every countable covering has a
point-finite refinement. In this paper, we･study some prope,rties of countably meta-
compact spaces. ･ The topological product Xx Y of a countably metacompact space X
and a compact space Y is countably metacompact. In order that a topological space
is countably metacompact it is necessary and suthcient that for e.very decreasing
sequence {Fi} of closed sets with empty intersection there is a sequence {Gi} of open
sets with ee
nGi--O such that Gi )Ei. There is a T2-space which is not countably
metacompact, and there is a countably metacompact and Regular T2-space which is
not countably paracompact.
1
Let X be a topological space, that is, a space with open sets such that the union
of any family of open sets is open and the intersection of any finite number'of open
'sets is open. A covering of X is a family of open sets whose union is XL The
covering is called countable if it consists of a countable family of open sets or finite
if it consists of a finite family of open sets. The lcovering is called 1ocally finite if
every point of X is contained in some open set which intersects only a finite number
･of sets of the covering and point-finite if each point of X is only a finite number of
sets of the covering. A covering 8 is called a refinement of a covering U if every
open sets of 8 is contained in some open set of U. The space X is called countably
paracompact if every countable covering has a locally finite refinement and countably
metacompact if every countable open covering has a point-finite refinement.
'
The purpose of this paper is to study the peoperties of countably metacompact
spaces. ･,
'
'
A space X i$ called compact if every covering has a finite refinement, paracompact
if every covering has a locally finite･ refinement, metacompact*** if every covering has
a point-finite refinement. It is 'clear ,that every compact, paracompact, or countably
paracompact space is countably metacompaet. Every closed subset of a countably
metacompact space is countably metacompact.
It is known that the topological product of a countably paracompact space X and
a compact space Y is countably paracornpact. We shall prove the similar theorem on
countably metacompact spaces.
* Department of Mataematics, Col!ege of General Education.
** We denote the empty set by O. ,
*** Cf. [1] p. !71. (Numbers in brackets refer to the references at the end of the paper.)
162 Y. HAyAsHi
Theorem 1. The toPological Product Xx Y of a countably metacomPact sPace X and
comPact space Yis countably metacomPact. '
[Proof] Let {Ui} (i--lj 2, ･･･) be a countable covering of XX Y.' Let Vi be the set
of all points x of X such that xxYCUUj. If xEVi every point (x,y) of xxY has a
js
neighbourhood NxM (N is an open set of .X; M is an open set of Y), which is contained in the open set H.iUj. A finite number of sets these open sets M cover Y; let
IVle be the intersection of the corresponding finite number of sets AIL The xEAZle, Alle is
an open set of X, and ATle×Y9Y.iUj, and hence IVleCVi. Therefore Vi iS an open set
of .XL AIso, for arfy xEX; sincexxYis compact, xxY is contained in some finite
goUvMebreinrgOoff SxtLtS Qf the COVering {Ui}; hence x is in some' Vi. Therefore,{v,} isa
Since {vi} is countable and x is countably metaEompact, {Vi} 'has a point-finite
refinement E!B. For each open set W of EEB let g(va) be the first Vi cotitaning VV and
let Gi be the union of all W for which g(W)=Vi. Then Gi is open, GiCVi and {Gi}
is a point-finite covering of X.
For each ]'s;i, set Gij=(GixY)nUj; then Gij is an open set in Xx Y. If (x, y)
is a point of (X) Y) then, for some i, xEGi and hence (x,y)EGxY. Also, since
xEGiCVi. (x, y)ExxYCjY?.,Uh and hence, for some 1'f{gi, (x,y)EUj. Hence (x,y)EGij.
Therefore {Gij} is a covering of Xx ]YL Since GijCUj, {Gij} is a refinement of {Ui}.
Also, if (x, y)EXx ]V; x is in only a finite number of sets of Gi. And hence, (x, y)g' is
in only a finite number of sets of {Gij]. Therefore Kx Y is countably metacompact. '
'
'
2
'
Theorein 2.* in order that a topological space X is countably metacompact it is
necessaf:y and szaficient that .ICbr every decreasing sequence {Flr} of closed sets with
fhMaet:GY ,/;ntRtr.SeCtiOn there iS a Sequence {Gi} of oPen sets with empty .{intersection such
A topological space X is called regular if for every pair (A,x) such that A is a
closed set.,and x is a point ￠A :there is a pair of disioint open sets U and V with
U )A and VEx.
A 'topolQgical space X is called normal if for every pair of disjoint closed sets A
ahd 'B there is a pair of disjoint open sets U and V with UDA and V)B. The
fellowing two properties of a normal space X are equivalent**:
(1) The space K is countably para,compact.
(2) The space Xis countably metacompact. '
However, these properties (1), (2) of some regular space are not equivalent, that is
the fol!owing theorem holds.
On Countably Mbtacompact SPaces 163
cTohuenOtr
ae ly p3bracTollf;laictZ.S
bM
a COUntably metacompact and reguiar space which is not
[Proof] Let Ri= {O, 1, 2, 3, ･･･, to, ･･･, 2} Where 2 is the first odinal number in all 3rd-
class ordinals, and R2={O,.1, 2. 3, ･.･, tu} where to is the first ordinal in all 2nd-class
ordinals. For each of Ri and R2,.we define its topology by the,limit of ordinals as
usual.* Set R=RixR2 and give the weak topology of the product space to R. And
let X==R-(2, tu). '
'
Since both Ri and R2 are regular, R is regular therefore .X is regular. We shall
prove X is not countably paracompact.
Let Fi={(9, n)ln=i, i+1,i+2. ･･･}(i=1,2, 3, ･"). Then {E} is a decreasing sequence
of closed sets of X wigh-empty intersection. Suppose that there is a sequence {Gn}' of
open sets of X with ,nn,Gi**=O such that GDE. Since (9, n) (n2i) is contained in
Gi, there is an ordinal ai,n<2 such that {(e,n)lai,n<es;;9} ]Gi. ' Set' ai=supai,n then
n {(a,n)i
ai<2 and {(e,n)Iai<e:{l;9, n= i, i+1, ･･･}CGi. Set a=supai then a<2 and
i
a<6f{;2, n==i,' i+1, ･･･}CGi therefore, for every at'(a<a'<9). (a', tu)cGi (i=1, 2, 3, ･･･).
This is contradictoty to the assumption nGi=O. Therefore, by a theorem in [3], X is
i
not countably paracompact.
We shall prove that X is countably metacompact,
Let {Fi} be a decreasing sequence of closed sets with empty intersection. Denote
Ri-coordinate of each point x of X by Ri[x] and R2-coordinate of x by R2[x]. We
shall prove that there is an integer m such that Ri[x]=2 VxE,Pla. Suppose there iS
some xEEn with Ri[x] =2 for･each m(lf{l;m<oo). Let xi (i=1, 2, ･･･) be a point of Fi
with Ri[xi]42. SetB==lim***Ri[xi], then BL<2. And let {yjll'--1,2,･･･} be a subbco
sequence of {xi} whith limyj==B. Set l=limR2[yj] and let {2icIle==1,2,･･･} be a sub-
j-eeAsheo
seqdenc of [yj] with limR2[zic]=l.
B<2 and IS:tu, (B,l)EX And lim2ic=(B,l).
ic-).ee ic-oo
Therefore, since {R} is decreasing and {2ic} is a subsequence of {xi} and xiER
(i=1,2, ･･･), UE)(3,l), that is nR40. This is contradictory to the.assumption 'nR =O.
Therefore there is an integer m such that Ri[x] =9 VxejF;n. Let Ri[x]==9 VxE"F;n.
Since {Fi} is decreasing, Ri[x]=2 vxEFi. for every l)}lm. And also R2[x]<di VxEFL
for every l)}lm. We define Gi (i=1, 2, 3, ･･･) as follows:
(a) Gi=X (i=1, 2, ･･･, m-1),
(b) Gi-- {xl there is a point yEFI with R2[y]==R2[x]} (i----m, m+1, ･･･).
Then Gi (i=1,2,･･･) is open and GiDE. We shall prove that fiGi=O. Suppose nGi=O
tt =R2[x]
and xEfiGi. From the definition, there are ym, ym+i, ･･･ such that yiEF; and R2[yi]
i
(i=m, m+1, ･･･). As .Ri[yi]==9 (i=mm+1, ･･･),y.=ym+i=･･･. Hence ?F}=¥O. This
* We define neighbourhood of a point p as follow$: for each q<p, {p'lq<p'E{;p} is a
neighbourhood of p.
** A is the closure of A.
*** lim is the upper limit of ordinals.
164 Y. HAyAsfii
is contradictory to the assumption nn=:O. Therefore we have nGi--O. Hence, by
ii
Theorem 2, X is countably metacompact.
Theorem 4. There is a T2-space which is not countably metacomPact.
[rroof] Set A :{(eir)I Os{;e<2, Of{;rE{:1} where 9 is the first ordinal number in all
3rd-class ordinals and r is a real number. And set B=={PIO<P<1, P is a rational
'
number}. '
Let X=AUB, and give the topology to X as follows:
X(a) If xEA, the set {x} which consists of only a point x is a neighbourhood of x.
(b) If xEB, for every B (Of{g;B<9) and every real numbere (O<x-e, x+e<1)
{(g, r)EAIB<e<2, x-e<r<x+e} u {x} is a neighbourhood of x.
Then X is a T2-space.
As B is countable, we may assume that B :{Pi, P2, "', Ps,} (s<ol)･
Let R={Pjl7'>i} (i---1, 2, 3. ･･･), then each E･ is closed and {]P?Ii=1, 2, 3, ･･･} is
decreasing sequence with empty intersection. We shall prove that there is no sequence
{Gi} of open sets with empty intersection such that Gi[Fi. Let {Gi} be a sequence
such that Gi),F- Since GiDFi, for each 1':;}li, there are some ordinal ai(1')<ve and
some real number ri(j') (O:f;Zri(]')<1) such that {(6, r)EAIai(1')<6<9, Pi-ri(1')<r
<Pj+ri(7')}CGi. Set `ti----1,j.m.co*ai(1') then cri<2and {(e,r)EAIcui<e<2, Pj-ri(7')<r<PJ
+ri(j)}(IZGi for each 1'2}li. Let AE=:{(e,r)EAIO<r<1} for each e<2 and let C[e]
==
{rlac}f'(a', r)ECAAe ifCCX. Then, from the above argument, Gi[e])Ei=U(Pj-ri(i),
j)t .Ei is an
Ri+ri(7'))** for every e>ai. Since {Pjl7'=i, i+1, ･･･} is dence in [O, 1]***
open set which is dence in [O, 1]. Set ev=:limai, then Gi[8]]Ei for every 8<a, hence
i-).oe
CGi[g]]9. Ei for every e>ev. Since each
Ei is an open set which is dence in [O, 1],
hence. by Baire's theorem, nEi is dence in [O, 1],that is UEi is not empty. Therefore,
ii
for every e>ev, OGi[e] is not empty. Hence. from the definition nGi is not empty.
Therefore there is no sequence {Gi} of open sets with empty intersection such that
Gi]Fi. Hence, by Theorem 2, the space X is not countably metacompaet. This completes the proof.
The space X in the proof of Theorem 3 is a regular T2-space. Theref6re, there is
a countably metacompact and regular T2-space which is not countably paracompact.
Also there is a T2-space which is not countably metacompact. However, the auther
knows no example of a regular T2-space which is not countably metacompact.
References
[1] J.L. Kelley: General Topology (1955).
[2] C.H. Dowker: On countably paracompact spaces, Canadian Jour. Math., 3, 219-224 (1955).
[3] Y. Hayashi: On countably paracompact spaces, Bull. Univ. Osaka Pref., Ser. A 7, 181-18'3
(l959).