The independence of certain axioms of structures in sets

Abstract:

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 $\emptyset$ as a fixed set and two covering properties ($\alpha$-character, with $\alpha$ 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 number $\alpha$.