Some generalizations of collectionwise normality

奈良教育大学学術リポジトリNEAR

Some generalizations of collectionwise normality

著者 SUZUKI Jingoro

journal or

publication title

奈良学芸大学紀要. 自然科学

volume 14

page range 1‑6

year 1966‑02‑28

URL http://hdl.handle.net/10105/3336

si?s* (iff.) SBi4@ aaw4i^

J. Nara Gakugei Univ. (Nat.), Vol. 14, 1966

Some generalizations of collectionwise normality

Jingoro SUZUKI

(Department of Mathematics, Nara Gakugei University) (Received Sept. 29, 1965)

§ 1. In a topological space X we consider the following two relations:

(d) For any discrete collection {Aa | aeT} of closed subsets of X there exists a sequence {Ui-{Uia\ a<^F}\ 1=1,2, } of mutually exclusive collections of open subsets of X such that for any element a. of F Aa is contained in w JJia.

(C2) For any discrete collection {Aa | ffiG/"} of closed subsets of X there exists a sequence {&i={Uia j «G F) | i=l,2, } of discrete collect-

ions of open subsets of X such that for any element a of F Ax is

contained in ^JJia-

We shall call a topological space X satisfying the relation (Q) a Ci-space (i=

1,2). In this note we shall investigate some propeties of Cr-spaces (i=l,2).

We shall use the following definitions:

Discrete. A collection of point sets is discrete if the closures of these point sets are mutually exclusive and any subcollection of these closures has a closed sum.

Screenable. A space is screenable if for each open covering ll of the space, there is a sequence Ux, U2, such that U4 is a collection of mutually exclusive open subsets and J^U* is a covering of the space which is a refinement of tt.

A space is strongly screenabie if there exist such II/s which are discrete collecti- ons.

Collectionwise normal. A space is collectionwise normal if for each discrete collection SI of point sets, there is a collection ll of mutually exclusive open subsets covering SI* such that no element of ll intersects two elements of 31, where §1*

is the sum of the elements in SI.

Jingoro Suzuki

Development. A sequence XLlt U2, of open coverings of the space is called development if for each point p and each open set U containing p there is an integer n such that every element of UM containing p is a subset of U. A deve- lopable space is a topological space that has a development.

Moore space. A Moore space is a regular developable space.

Remark 1. The property (CO is a generalization of that of collectionwise normality (t=l,2). For the normality we can consider the similar generalizations as follows:

(N,) For any disjoint closed subsets A and B of a topological space X there existsa sequence {Ut \ i-1,2, } of open subsets of X such that A is contained in ^Ui and Uir~,B=<j> for any integer i.

(N2) For any disjoint closed subsets A and B of a topological space X there

exist two sequences {Ui\ i=l,2, } and {Vi\i-1,2, } of open

subsets of X such that A (B) is contained in ^JJi (>r^0 and Uir~sVl

=<j> for any integer i.

We can easily prove that a space with the property (NO is normal aud any space X satisfies the property (N2).

Remark 2. Any closed subsets of a Crspace is a C-space (i=l,2).

§2. From the definitions property (C2) implies property (CO. In §2 we shall show that property (CO is weaker than property (C2) in general, and some other related properties will be stated.

Example 1. A Hausdorff, C2-space need not be a regular space, in

general. Let X be a subset {(x,y~) \ x,y-. rationals, y^o} of a two-dimensional Euclidean space R2. The set X is topologized as follows. For any point of {(x,j)eX jy>o} the nbd is defined as usual in R. For a point (<2,0) el we define the nbd of (a,o) by the intersection of a spherical nbd of (c,o) in R with {{x,y)^X | y>o}^ {(c,o)}.

We shall show that X thus topologized is a C2 -space. Suppose a sequense {Pi,p2, } is a well ordering of X. Let {Aa \ aer} be a discrete collection of closed subsets ofX. For any pf and a^T we define a open set Uia such that

Uia=<j> if Pi $ Aa and Uia is a nbd ofpt if/»,gA,. Then the sequence {VLla={Uia ccgT} I 1=1,2, } of discrete collections of open subsets of X satisfies the property (C2).

Remark 3. From the discussion of Example 3, we know that a topological space consisting of countable points is a C2-space.

Example 2. Regular, Ci-space that is neither normal nor C2. Example

Some generalizations of collectionwise normality 3

B of Bing (j[~) is suitable for this example. The space which appears in the exa- mple is a screenable non-normal Moore space. A screenable space implies a Ci-space by our Theoem 3 of §3. If the space is C2-space, then the space would be metrizable by our Corollary of Theorem 7. This contradicts non-norma- lity. Hence, the space is not a C2-space.

Example 3. A normal space that is not a Cx-space. For this example we shall consider the Example G of Bing Qj: Let P be an uncountable set, Q the set of all subsets ofP, and X the set of all functionsfon Q having only 1 and 0 as values. To each element p of P associate the function fp whose values fP(_q) is 1 ox 0 according aspbelongs toq ornot. Let Fp be the set of all such

functions fp. The set X is topologized as follows. Any point / in X-Fp is declared to be nbd of itself. Given a pointfp in Fp and a finite subset r of Q we define the r nbd offp to be the set of all / such that f(q)=fp(q) whenever q belongs to r.

If we take the discrete collection {{fp } p^P} of closed subsets of X, then wecan not construct any sequence which satisfies the condition of (Cj). It is

proved as similar as Bing's proof of non-collectionwise normality.

Now, we shall prove the following two theorems.

Theorem 1. In order that a C2-space X be collectionwise normal, it is necessary and sufficient that for any discrete collection {Aa j a^r) of closed subsets of X there exists a sequence {ll4} which satisfies the condition (C2) and U,X/^Aa=4> for any i, a, P (a=^/3).

Proof. Since the necessity is evident, we shall prove the sufficiency. Put Mia=

pV* Utft. Then Mia is closed and Miai^Aa=</>. Let us put Vna=Una - £ Mia for any n,a and Va=^[Vna, for any a<=Ft then we have a mutually exclusive collect:o:i {Va} of open subsets of X such as ^cy^for aeT. Hence, X is collectionwise normal.

Corollary. A normal C2-space is collectionwise normal.

Theorem 2. Let X be a C2-space. Then for any locally finite collection {Fa a<E.r) of closed subsets of X whose order is finite there exists a sequense {{Ula, aeT} i=l,2 } of locally finite collections of open subsets of X

such that w Um'OiF*.

Proof. We assume that the order of {FJ is equal to n. We shall prove the

theorem by induction of n. If n=l, the theorem evidently hold. Assuming that the

Jingoro SUZUKI

theorem has been established for any n^Lm, we shall prove for n=m+1. Let us put %~{a(zr j thepower of a is ejual to m+1}. Then {^aFa j aeSt} is a discrete collection of closed subsets of X. Hence, there is a sequence { {Uli1 J

#£=21} ( i=l,2, } of discrate colleotions of open subsets of X such that >-^

U1^1 ID ^ K for any ae2l and such that for any i Z/%1 ^ Fa±?? if and only if a^a. Let us put U= ^ U"1^1, then the order of the locally finite coll- ections {Fa-U\ ae^r} of closed subsets of.Xdoss not exceed m. Consequently, there exists a sequence {{Vice ) oiEzF} | {=1,2, } of locally finite collections of open subsets of X such that ^ Via ID Fa-U{or any ajeT. Finally, let us

put Uix=Via ^ (^ Umil). Then the collections {{Uia ) aeT} \i=l,2, >

satisfies the conditions of the theorem.

§3. In this section we shall consider some relations between screenable (strongly screenable) spaces and C^Cz)-spaces.

Theorem 3. A screenable {strongly screenable) space X is a d(C2)-space.

Proof. Let {A& aeT} bs a discrete colletion of closed subsets of X. For any pointp of Ax we take a open nbd Up# ofp such that Upa ^ (££,Ap)=</>.

Then there exists a sequence {%$t} of mutually exclusive (discrete) collections of open subsets of X such that ^ 334 is an open covering of X and w 33S is a refi-

nement of {X-^AX}^{UVX \P^Aa, a^r}. Put Uu, = {V\ V^Aa ^^,Fe

SS{}. Then {Mi={Uu j aeT} j i-1,2, } is a sequence of mutually exclusive (discrete) collections of open subsets of X which satisfies the condition (CL) ((C2)).

Theorem 4. Let X be a pointwiseparacompact space. Then Xis a d-space if and only if it is screenable.

Proof. The sufficiency is proved by Theorem 4. We shall prove the necessity.

Let U be an arbitrary open covering of X and Bi a point finite open covering which refines U. Let Mi be a set of all points which belong exactly to one element of &. Put ^^{V^M, \ V^flf,=j5, Fe^}. Then S3, is a discrete collection of closed subsets of X. We can construct a sequence {ViJ = {£/£ ! ojeTi} J {=1,2, } of mutually exclusive collections of open subsets of X

such that {Uu' ] i= 1,2, } satisfies the condition (d) for the collection SS^ For any V^MxGE53iwe can select an index a^T1 such that >-^U}a cV^ Mi by the

condition (Ci). If we put V}a = Ul/-N Ffor anyiand a, then {UH={Vl\a<=

Fij ) i=l,2, } is a sequenence of mutually exclusive collections of open

subsets ofXand Y« Via zd Mx.

Some generalizations of collectionwise normality

Now, letus put ^={V1 ^V2\ Vu ViEffl!, V1^V2,V1 ^V2 =^}^ {£UU>

and M2 a set of all points which belong exactly to one element of l?2. Then,

%>2={V1^V2^(M2- C H* )^ j V^V^ftz } is a discrete collection of closed subsets of X. We can construct a sequence {tT2i={t/£ ffE^} | i-1,2, } of mutually exclusive collections of open subsets of X such that {W2i \ i=l,2, } satisfies the condition(Ci) for collection 932. For any Vi^V2^(M2-^ Uft )g$2 we can select an index aeT, such that ^ U*, 3 Fi^F^C/l^-å ^1IJ ) by the

condition (d). We shall put V£=f/^F.^F,,. Then, {H2i= {F£ I a<=T2 } \ 1=1,2, } is a sequence of mutually exclusive collections of open subsets of X such that (^U* )wQr^2* ) contains the set of all points which are contained in at most two elements of ^x and any element of (^Hh )^(^> U2I) is contained in some element ofBlm By the same process, for any integer n we can construct a sequence {1IMJ | i=l,2, } of mutually exclusive collections of open subsets of X such tnat (^tt* )w w(^U*f ) contains the set of all points which are contained in at most n elements of ^ and any element of ^ C- Uji is contained in some element of $i. The sequence {VLni j n,i=1,2, } of mutually exclusive collections of open subsets of X satisfies the conditions of screenability for the open covering ll.

Theorem 5. Let Xbe apointwise paracompact spa.ee, then Xis a C2-space if and only if it is strongly screenable.

Corollary. If X is a regular, pointwise paracompact, C2-space, then X is paracompact.

Theorem 6. Let Xbe a developable space, Then X is a Ci-space if and only if X is screenable.

Proof. Since the sufficiency is clear, we shall prove the necessity. Let ll be

an open covering of X and {Gt} a development of X such that for any integer

i d is a refinement of ll. By the Theorem 9 of Bing \Jl} there exists a sequence

{%i} of discrete collections of closed subsets of X such that for any element F

of §f4 S(F,G{^) is contained in some element U ofll and >^ $t is a closed covering

of X. Hence, for any integer i there exists a sequence {VLtj j j-1,2, } of

mutually exclusive collections of open subsets of X such that for the collection §4 the

condition (Ci) is satisfied and any element of VUj intersects just one element of %i.

6 Jingoro Suzuki

For any Fegt and integerj let us put VlF} = S(F, 1X^)^11, where Uis an element of U which contains S(,F,Gt~). Then {58u={y#j F^%t} \ ij=l,2, } is a sequence of mutually exclusive collections of open subsets of X such that Yj^ij is a covering of X and ^^ij is a refinement of ll. Consequently, X is screenable.

Theorem 7. A necessary and sufficient condition that a developable space X be a C2-space is that X be strongly screenable.

Corollary. A Moore, C2-space is metrizable.

References

(1) R. H. Bing, Metrization of topological spaces, Canadian Jour. Math., 3 (1951),pp.175-186.

(2) W. Heath, Screenability, pointwise paracompactness, and metrization of Moore spaces, Canadian Jour. Math., 16 (1964), pp. 763-770.

(3) M. Katetov, Extension of locally finite coverings, Colloq. Matn., 6 (1958), pp. 141-151.

(4) R. L. Moore, Foundations of point set theory, Am. Math. Soc. Coll. Pub., 13 (1932).