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This article is a continuation of study of star-Menger selection properties in line of (Kocinac, 2009, 2015), where we mainly use covers consisting of G? sets with certain additional condition. It is observed that star-Mengerness is equivalent to the fact that every such type of cover of a space has a countable subcover. We improve this result by considering ?subcovers of cardinality less than b? instead of ?countable subcovers?, which is our primary observation. We also show that it is possible to produce non normal spaces using box products and dense star-Menger subspaces.

In this paper we primarily introduce the local version of star-Menger property, namely locally star-Menger property (a space with this property is called locally star-Menger) and present some important topological observations. Certain interactions between the new notion and star-Menger property are also observed. Some observations on effectively locally star-Menger Pixley-Roy hyperspaces (introduced here) are obtained. Preservation like properties under several topological operations are also interpreted carefully. Besides, several results on decomposition and remainder of locally star-Menger spaces are also presented.

We intend to localize the selection principles in uniform spaces (Kocinac, 2003) by introducing their local variations, namely locally ?-bounded spaces (where ? is Menger, Hurewicz or Rothberger). It has been observed that the difference between uniform selection principles and the corresponding local correlatives as introduced here is reasonable enough to discuss about these new notions. Certain observations using the critical cardinals (on the uniform selection principles which have not studied before) as well as preservation like properties (on the local versions) are presented. The interrelationships between the notions considered in this paper are outlined into an implication diagram. Certain interactions between these local variations are also investigated. We present several examples to illustrate the distinguishable behaviour of the new notions.

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