SOME RESULTS ON/n,m]-PARACOMPACT AND/n,m]-COMPACT SPACES

. Let n and rn be infinite cardinals with n < rn and n be a regular cardinal. We prove certain implications of [n,m]-strongly paracompact, [n,m]-paracompact and [n,m]-metacompact spaces Let X be In, oo]-compact and Y be a [n, m]-paracompact (resp. [n, oo]-paracompact), P,-space (resp. wP,-space). If rn m k we prove that X x Y is [n, m]-paracompact (resp In, c]- k<n paracompact)


INTRODUCTION
Throughout this paper rn and n will denote infinite cardinals with n < m and n will be a regular cardinal. A space X is called [n, m]-compact (see Alexandroff [1]) if every open cover o of X with Icl < m has a subcover of cardinality < n. For a set A, we denote by IAI, the cardinality of A. A family of subsets of X is a locally-n (point-n) family (Mansfield [2]) if for every x 6 X, there is an open neighborhood of x in X which meets < n members of c (resp x ,belongs to < n members of a) An open refinement of a cover a of a space X is an open cover/ such that each member of B is contained in some member of c. A space X is [n, m]-paracompact (resp. [n, m]-metacompact) if every open cover of X with I1 < m has a locally-n (resp. point-n) open refinement. X is [n, m]-strongly paracompact if every open cover of X with Ic, <_ m, has an open refinement such that for each Bs#, Originally, Singal and Singal introduced the concept of (m, k)-paracompactness in [3]. Our notation is slightly different than theirs. However, we note that a space X is (m, k)-paracompact, as defined in [3], if and only if X is [k+,m] [7] that if X is a locally compact Hausdorff space, then X is paracompact if and only if X is a disjoint topological sum of a-compact spaces. It is natural to ask when X is a locally In, oo]-compact, [n, oo]-paracompact space, whether X is a disjoint topological sum of a-[n, oo]compact spaces. The result above is the answer to the case when n w0 and X is Hausdorff. So we are only interested in the case when n > too. The following theorem provides the answer to this question TIIEOREM 2.5. Let n > w0 and X be a locally [n, oo]-compact regular space. Then X is In, oo]paracompact if and only if X is a disjoint topological sum of In, oo]-compa'ct spaces PROOF. It is obvious that if X is a disjoint topological sum of [n, oo]-compact spaces, then X is [n, oo]-paracompact. Thus let us assume that X is [n, oo]-paracompact Let c {U" U c_ X and U is In, oo]-compact}.
Then/ {int U U a} is an open cover of X since X is locally [n, oo]-compact Since X is regular, then there is an open cover 7 of X such that q {c eG G e 7} refines . Since X is a locally [n, oo]compact, In, ]-paracompact space, then by Theorem 2.3, X is In, oo]-strongly paracompact Hence there exists an open refinement a of 7 such that for each L a the set/XL {H cr L H -} has cardinality n. For a positive integer t, a chain of length t in a is a sequence L1,..., Lt in a such that L A L,+I for 1 < < t 1. If 1 we simply require L1 : . For x, y E X we define x y if there is a chain L1,..., Lt in a such that x L1 and y Lt. Clearly is an equivalence relation since a is an open cover of X. Let R be an equivalence class and a R. If y E R, then there is a chain L1,..., L in a such that a E L1 and y ELt. Clearly each point in Lt is equivalent to a with respect to ", hence Lt C R. So R is open. Let z E ceR. There exists L E a such that z E L Since z ceR, then L fq R :/: Thus if w E L n R, then z w, i.e, z E R. This shows that R is also closed Let a E L and L E a. We know that L C_ R For a positive integer t, let there is a chain L1, Lt in r such that L L1 and Lt H}. when is a positive integer, then Rt is also In, oo]-compact Since n > w0, then R to P is also In, oo]-compact. This proves the theorem since X is the disjoint topological sum of the equivalence classes of ".

PRODUCT THEOREMS
In this section we prove theorems concerning [, ]-paracompact of a product space X Y Our first theorem is a generalization of a result by Morita [5] which states that if X is a compact space and is an -paracompact space, then X Y is an -paracompact space.
THEOREM 3.1. Let the cardinal rn satisfy rn l{rn k k is a cardinal and k < n} Let X be an [n, oo]-compact space and Y be an [n, m]-paracompact P,-space. Then  IMI which is less than n since I'1 < n, IMI < n for each M E #' ME/.t' and n is a regular cardinal Hence p is a locally-n family In Theorem 3.1 if we assume the stronger condition that Y is In, oo]-paracompact then we can show that X Y" is [n, oo]-paracompact if we only assume that is a wP,-space Before we prove this result we first prove two theorems which are interesting in their own rights Let A and/3 be topological spaces and f" A --o B be a function f is called n-closed if for every closed subset F of A, f(F) is an n-closed subset of B. PROOF. We will only prove the case when Y is In, oo]-paracompact. The In, oo]-compact case can be proved similarly.
Let a be an open cover of X For each i/ Y let a u be a subcollection of a such that I,1 < r and f-l(/) c_ L.ja u. Such a subcollection exists since f-l(!/) is [n, oo]-compact. For Y, let G U a u, and W u Y\f(X\Gu). Then i/ W and W u is n-open since f is an n-closed map Thus for each / Y, there is an open set Vu in Y such that /E V and Vu\Wu] < n. 7 { Vu U Y} is an open cover of Y and Y is In, oo]-paracompact. Hence there existg a locally-n open refinement {T I} of 7. For each I, pick / Y such that T, C_ V,. For / Y let u u (,, .t v\w)).
Then II <_ I,I + (I,I t v\w) < , since n is a regular cardinal. Moreover f-1 (Ti) C_ U/, since T, C_ V,. Let Then clearly a is an open refinement of c. Let :r X and i/= f(z). There is an open set N in Y and a subset 3 of I such that Jl < n, N and N n T for all /3. Let M f-(N) and A { H l-! f-(T) H /u,, 3}. Then z E M and IAI _< Itu, < n since n is a regular cardinal Moreover, if L E or\A, then L r3 M or. Hence cr is a locally-n family.
As a corollary of Theorem 3 2 and Theorem 3 3 we obtain the following variation of Theorem 3