The intersection multiplicity of $n$-dimensional paracompact spaces

Abstract

It is shown that there is an integer v(n) < 32n+1 such that any n-dimensional paracompact space X has intersection multiplicity at most v(n). That is, if Ii is an open cover of X, then there is an open cover (9 refining iJ such that any element of t9 intersects at most v(n) elements of O. For any open cover CU of a topological space X define m('U) to be the maximum number of elements of cU that any member of cU can intersect. X is said to have intersection multiplicity at most m if every open cover CU of X has an open refinement (9 covering X such that m(Q) < m. The intersection multiplicity of X is then the least integer m such that X has intersection multiplicity at most m and is denoted m(X). Intersection multiplicity is clearly preserved by homeomorphisms. Also if A is a closed subset of X, then m(A) < m(X)X Received by the editors February 15, 1974 and, in revised form, May 7, 1974. AMS (MOS) subject classifications (1970). Primary 57C05, 54D20, 54F45, 55C10.