Let P:2v->-2v be a structure in a topological space V such that P(0 )=0 , P({x})={x] if x e V, and P(Z) is closed if Zç V. If G is a covering of V, let Gz={Xe G:xe X}. If A' is a set and y is a set, let [X\ denote the cardinal number of X and X— Y= {x€X:x(fc Y}. Let n be an integer such that n^. — 1. dimP V is defined as follows: dinv K= -1 if V=0 . If Vjt 0 , then dim,, V= n if (1) for each finite open covering G of V, there is an open refinement Hof G such that \HX\ ̂/i +1 if x e V; and (2) there is a finite open covering G of V such that if H is an open refinement of G, then \HX\ ̂« + 1 for some x e V. We say that P has property (*) if for each nonempty open T<= V and each X^ V such that P(X) ?¿ V and x $ P(X-{¿-»whenever*e JTandeachxe[V-P(X)], [ Y-P{X)]r\ P(Xkj{x})^ 0. Theorem. If V is a metric space, P has property (•), B£ V, B is finite,P(B)= Vandx $ P(B-{x}) ifxeB, then dinv K= 1*1-1.

https://www.ams.org/proc/1973-041-01/S0002-9939-1973-0322828-2/S0002-9939-1973-0322828-2.pdf

Covering dimension in finite-dimensional metric spaces

The class of all combinatorial geometries of rank three shall coincide with the class of all pairs (V, S) such that V is a set and S is a collection of non-empty subsets of V such that each pair of distinct elements of V belong to exactly one member of S. (See [3].) Consider a combinatorial geometry (V, S) of rank three.

Independence in Combinatorial Geometries of Rank Three

The independence of the axioms for spans and the independence of the axioms for closure structures are usually taken for granted. In this paper, the author establishes the independence of monotonicity, extensiveness, idempotence, the exchange property, the property of having 0 as a fixed set and two covering properties (cc-character, with ac being some cardinal number, and a covering property with respect to generators). The independence of the axioms for closure structures and spans follow immediately. It is shown that any proof of the independence of a given axiom must involve an example with certain restrictions on the cardinal

/pdf/the-independence-of-certain-axioms-of-structures-in-sets-4j6dhm52cg.pdf

Japheth Hall

Papers

The independence of certain axioms of structures in sets