A remark on irreducible spaces

Abstract

Abstract.A topological space X is called irreducible if every open cover of X has an open refinement which covers X minimally. In this paper we show that weak 0-refinable spaces are irreducible. A modification of the proof of this result then yields that « [-compact, weak 50-refinable J. Boone gave a proof for the above result of Worrell and Wicke and then extended this technique using an involved argument to show that weak 6-refinable spaces, introduced by the author in [11], are irreducible.In §2 of this paper we investigate the conditions under which certain Fasubsets of a topological space have minimal open covers. As an application of this we provide a relatively simple constructive proof of Boone's second result.In §3 we give conditions which ensure that weak 50-refinable and weak 88-refinable spaces are Lindelöf and irreducible. In particular, we show that incompact, weak 00-refinable spaces are Lindelöf. Various open questions are