Weaker Forms of the Menger Property

In this paper we study a weaker form of the classical concept of Menger property. This property, called weakly Menger, is independent from the Menger property and the almost Menger property. In particular, for Tychonoff spaces weaker Menger spaces are not equivalent to almost Menger spaces. We give some characterizations in terms of regular open sets and almost continuous mappings. We also introduce a weaker form of the star-Menger property and the notion of almost gamma k-set. Some open questions are given.