On m-paracompact spaces IIIn a recent paper [2], we have obtained some results on m-para-

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V II (1973) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATE MAT YCZNE X V II (1973)

M. K.

and

(Delhi, India) On m-paracompact spaces II

In a recent paper [2], we have obtained some results on m-para- compaet spaces. In the present note, we obtain some characterizations of m-paracompact spaces and show that the corresponding known results for paracompact spaces obtained by Tamano [4], Yaughan [5], Michael [1] and authors [3] follow as corollaries to our results.

De f i n i t i o n s.

Let s3 and 38 be two families of subsets of a space X.

зЗ is said to be cushioned in 38 if there exists a cushion map /: s3 -> 38 such that for every subfamily лЗ’ of лЗ we have

U {A': X'e лЗ'}<= U { / № ) : A' * О

лЗ is said to be linearly-cushioned in 38 with cushion map f: лЗ -> 38 if there is a linear ordering '< ' on лЗ such that for every subfamily лЗ' of ( л З, < ) for which there exists an X e лЗ such that X <c X for all X елЗ

we have

U{M': X'e лЗ'} я (J {f(X'): X'e s3'j.

лЗ is said to be order cushioned in 38 with cushion map /: лЗ -» 38 if there exists a well ordering '<' such that for every subfamily s3' of лЗ and Xe лЗ such that X' < X for all X'e лЗ' we have

C U C U { X ' n A : А ' е л З ' } ] я U { / ( ^ ' ) : Х' елЗ' } .

The concept of cushioned families was introduced by Michael [1]. The notion of linearly cushioned families is due to Tamano [4], who assumed the ordering to be a well ordering. However, it was proved by Yaughan [5] that if 'лЗ is linearly cushioned in 38 with respect to a linear order, then there also exists a well ordering with respect to which лЗ is linearly-cu­

shioned in 38.

The concept of order cushioned families has been introduced by the authors [3].

Th e o r e m.

For a space X, the following are equivalent:

(a) Y is normal and m-paracompact.

216 M. K. S i n g a l a n d S. P . A r y a

(b) Every open covering of X of cardinality < m has an open, cushioned refinement.

(c) Every open covering of X of cardinality < m has a o-cushioned open refinement.

(d) Every open covering of X of cardinality < m has a linearly-cushioned, open refinement.

(e) Every open covering of X of cardinality < m has an order cushioned, open refinement.

(f) Every open covering of X of cardinality < m has a cushioned re­

finement.

Proof, (a) => (b). Let °U = {TJX: Ae Л) be any open covering of X of cardinality < m. Since X is nt-paracompact, there exists a locally - finite, open refinement 'f' — {Vx: Ae Л} of °и. Since X is normal there exists a locally-finite, open refinement У — { Wx: Ae Л} of У such that Wx я Vx for each Ae Л. Then {Wx: Ae Л} is an open cushioned refinement of W.

(b) => (c). Obvious.

(c) => (d). Let be any open covering of X of cardinality < m.

oo

Let У = U be a a-cushioned, open refinement of so that each

n = l

iTn is cushioned in °ll. Let each i rn be well ordered and define an order '< ' in У as follows: if Уг, F 2e У , then Y\ < F 2 if (i) Уг < F 2 in the order­

ing of У n in case both Vx, У ^ У п (it being assumed without any loss of generality that different У п^ do not have any members in common) and (ii) Vx < V 2 if У \* У п-> and n < m. With this ordering, it is easy to verify that У is a linearly-cushioned, open refinement of °U.

(d) => (e). Let °U be any open covering of X of cardinality < m.

Then, there exists a linearly-cushioned, open refinement У of Qi with cushion map f: У -»■ °U. It can be assumed that the ordering of У is a well-ordering (cf. Vaughan [5], Theorem 1). We shall show that У is order cushioned in

°U. Let У be any subfamily of У and let У e У be such that У < V for all У . ' е У . Then ClF[ U { ^ ' n ^: У ' е У } ] = C1FI((J{^'-* Г ' е У } ) п Г ]

= U { V ' - У ' с У } n V n V я U { V ' : У ' е У ] г л У я \ J { V - . У е У } Я U {f(V'): Г ' е У } .

Hence У is an order cushioned refinement of °U.

(e) => (f). Let °U be any open covering of X of cardinality < m.

Let У be an open, order cushioned refinement of % with cushion map /: У We shall show that there exists a cushioned refinement of °U.

Let W v = V ~ V J { y • У < У} for each Fe У. Let У = {W v : Уе У} .

Let g: У be defined as д{УУу) = / ( F ) for each TУу е У . It

will be shown that У is cushioned in with cushion map g. First, let x e X.

m-paracompact spaces I I

Then there exists a smallest V e ' V such that x e V. Obviously then x e W v .

So I V is a covering of X each member of which is clearly contained in some member of Let IV' be any subfamily of I V . We shall show that U { W r : W V€ l V } я U { y ( W v ): W y t I V ' } . Let ye \ J { W r : W r e ! V ' } . There exists V e V such that y e V. Let M be any other open set containing y.

Then M n V is an open set containing y. Also, since V n F r = 0 for all V > V, therefore ( M n F ) n [ U { W v>: V < V, W v,e TF'}] Ф 0 and thus y e VJ{Wv- V ' < V v, , W , € W } . Let I V " = { W v,: V' < V , W v. e I V }

and let nV " = { V ' e V : W v>eIV"}. Since V < V for all V ' e i V " , therefore V" is majorized and hence we have

C lp[(U {F': V ' < У i V e V " } ) n F ] = Clr [ U { V n У : У с Г " } ] '

и

£ W v.e!V'}.

Now, since V is an open set, therefore we have C1

[(U { F ': 7 ' « Г } )

7 ] = y {F': 7 ' е Г } п 7 .

But ye U {Wy, : W v. e IV"} я U { V : 7 ' е Г } and also ye V. Therefore, 2/eClF[(U { F ': У е Г " } ) ^ я \ J { g { W r ,■): Wv. e!V'}.

Thus, U { Wv : W r e IV'} я (J {g(Wv.) : W v. e IV This shows that IV is a cushioned refinement of °U.

(f) => (a). This follows in view of a theorem obtained by authors (cf. [2], Theorem 1.10).

From the above theorem follows the following important corollary:

Co r o l l a r y

1. For a space X, the following are equivalent : (a) X is normal and paracompact.

(b) Every open covering of X has an open, cushioned refinement.

(c) Every open covering of X has an open, o-cushioned refinement.

(d) Every open covering of X has a linearly-cushioned, open refinement.

(e) Every open covering of X has an order cushioned, open refinement.

(f) Every open covering of X has a cushioned refinement.

In view of the fact that every paracompact, regular space is normal, we have the following known results which follow from the above corollary :

Co r o l l a r y 2

(Michael [1]). A regular space is paracompact iff every open covering has a cushioned (or open, o-cushioned) refinement.

Co r o l l a r y 3

(Tamano

A completely-regular space is paracompact

iff every open covering has a linearly-cushioned open refinement.

218 М. К. S i n g a l a n d S. P, A r y a

Co r o l l a r y

4 (Yaughan [5]). A regular space is paracompact iff every open covering has a linearly-cushioned, open refinement.

Co r o l l a r y

5 (Authors [3]). A regular space is paracompact iff every open covering has an order cushioned, open refinement.

References

[1] E. M ic h a e l, Y et another note on paracom pact spaces, Proc. Amer. Math. Soc.

10 (1959), p. 309-314.

[2] M. K. S in g a l and S h a s h i P r a b h a A r y a , On m-paracompact spaces, Math.

Ann. 181 (1969), p. 119-133.

[3] — Л note on order paracompaotness, Bull. Austral. Math. Soc. 4 (1971), p. 273- 279.

[4] H. T a m a n o , Note on paracompactness, J. Math. K yoto U niversity 3 (1963), p. 137-143.

[5] J. E. Y a u g h a n , L in ea rly ordered collections and paracompactness, Proc. Amer.

Math. Soc. 24 (1970), p. 186-192.

M E E R U T U N IV E R S IT Y , M E E R U T , IN D IA a n d

M A IT R E Y I C O L L E G E

N E T A J I N A G A R , N E W D E L H I, IN D IA