**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. **

**S**

**i n g a l**

**and **

**S**

**h a s h i**

**P**

**b a b h a**

**A**

**r y a**

**(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.**

**D****e f i n i t i o n s****. **

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

**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**

**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

**{A': X'e лЗ'}<=**

**{ / № ) : A' ***

**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 **

**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'**

**( л З,**

**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.**

**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**

**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**

**[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.**

**shioned in 38.**

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

**T****h e o r e m****. **

**For a space X, the following are equivalent:**

**For a space X, the following are equivalent:**

**(a) Y is normal and m-paracompact.**

**(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.**

**(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.**

**(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.**

**(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.**

**(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**

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

**finement.**

**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.**

**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.**

**(b) => (c). Obvious.**

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

**be any open covering of X of cardinality < m.**

**oo**

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

**Let У = U**

*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**

**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.**

**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. **

**(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 **

**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. Let**

**У**

**be any subfamily of У and let**

**У e**

**У be such that У < V for**

**У . ' е У .**

**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 } .**

**U { V ' - У ' с У } n V**

**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.**

**Hence У is an order cushioned refinement of °U.**

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

**(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 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 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 **

**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.**

**will be shown that У is cushioned in**

**with cushion map g. First, let x e X.**

*m-paracompact spaces I I*

217
**Then there exists a smallest ** **V e ' V** **such that ** **x e V.** **Obviously then ** **x e W v . **

**V e ' V**

**x e V.**

**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. **

**I V**

**X**

**IV'**

**I V**

**. We shall show that**

**{ W r : W V€ l V }**

**я U { y ( W v ):**

**W y t I V ' } .**

**\ J { W r**

**W r e ! V ' } .**

**V**

**V**

**y e V.**

**M**

**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 } **

**M**

**n V is an open set containing y. Also, since**

**V**

**V > V,**

**( M**

**{ W v>: V < V, W v,e**

**Ф**

**y e**

**VJ{Wv-**

**V ' < V v, , W , € W } .**

**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**

**nV "**

**{ V ' e V : W v>eIV"}.**

**V**

**V**

**V ' e i V " ,**

**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 " } ] '**

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

*Я*

** и **

*i f ( Г ) : V e r " } = \ J { g ( W y .) i W v .* iV " }*

**£ ** **W v.e!V'}.**

**W v.e!V'}.**

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

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

**f**

**[(U { F ': 7 ' « Г } )**

**п**

**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'}.**

**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.**

**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:**

**C****o r o l l a r y**

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

**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.**

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

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

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

**(e) Every open covering of X has an order 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.**

**(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 :**

**C****o 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.**

**(Michael [1]). A regular space is paracompact iff every**

**open covering has a cushioned (or open, o-cushioned) refinement.**

**C****o r o l l a r y**** 3 **

**(Tamano **

**[4]).**

**A completely-regular space is paracompact **

**A completely-regular space is paracompact**

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

**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.**

**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.**

**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