Addition and reduction theorems for medial properties

Abstract

In [6], I gave a systematic presentation of certain kinds of medial properties such as (P, Q)n, (P, Q, )n, etc., and their basic properties, having special regard for their dualities and relations to local properties. The present paper is supplementary to [6], in that it goes further into the relations between open sets and their complements and provides certain addition and reduction theorems not given in [6], as well as their applications. Most of the proofs depend upon chasing, and two diagram types recur frequently. Toavoid repetition, these types and their relevant properties are given in two lemmas in an Appendix. Throughout, point sets are assumed to be imbedded in a locally compact Hausdorff space X. Point set boundaries are denoted by the symbol F; thus F(A) denotes the boundary of the point set A. As in [6], a pair P, Q of open sets is called canonical if P- Q and Q is compact. Homology and cohomology groups based on compact supports are denoted by lower case h. As in [6], Cech homology and cohomology with coefficients in a field are used throughout. If is a subset of a space X, then a property of is called intrinsic if it is a topological invariant of A, and extrinsic if it is a positional invariant of in X (see [5, p. 290]). If is closed, the distinction is of no consequence, since the open subsets of coincide with its intersections with open subsets of X. But for not closed, the distinction is important-medial properties of in terms of its own open (rel. A) subsets are intrinsic, but in terms of the open subsets of X they are only extrinsic. (For example, the open set M of Example 1.1 of [6]-a domain in E2 bounded by a closed curve containing a sine curve of form y = sin 1 /x-has property (P, Q)o intrinsically but not extrinsically.) The expression A has property (P, Q)r will frequently be abbreviated to A has (P, Q), , and in similar expressions involving other medial properties. 1. Relations between open sets and their complements. In [6], the following question was considered: If M is closed and both X and M have certain medial properties, what can be concluded concerning the medial properties of X- M?