Neighbourhood Operators: Additivity, Idempotency and Convergence

Abstract

We define and discuss the notions of additivity and idempotency for neighbourhood and interior operators. We then propose an order-theoretic description of the notion of convergence that was introduced by D. Holgate and J. Šlapal with the help of these two properties. This will provide a rather convenient setting in which compactness and completeness can be studied via neighbourhood operators. We prove, among other things, a Frolík-type theorem with the introduction of reflecting morphisms. and Tholen [15] and Castellini [4] give a detailed account of the topic.