Proximity structures and grills

A note on mathematical structures

“…A mathematical tool grill [2,17,16] has been introduced for the purpose of the study of proximity spaces. A nonempty subcollection G ⊂ 2 X on a set X is called a grill on X if ∅ / ∈ G, A ∈ G and A ⊂ B implies B ∈ G and A, B ⊆ X and A ∪ B ∈ G implies A ∈ G or B ∈ G. The collection of all grills on a set X is denoted as G X .…”

“…A nonempty subfamily F ⊂ 2 X is called a stack [6,16] on X if ∅ / ∈ S and A ⊆ B, A ∈ S implies B ∈ S. The collection of all stacks on X is denoted as S X .…”

On a Problem of F. Riesz Concerning Proximity Structures

“…A more extensive discussion of the ideas employed here is given in a forthcoming article [8] by the author.…”

On the product of 𝑓-proximities

“…Generalized proximities have been extensively investigated by Lodato [9], Harris [6], Gagrat and Naimpally [5], Sharma and Naimpally [13], Thron and Warren [17,18] and others. Thron [16] introduced the concept of/-proximities and has shown that for different choices of /one obtains many of the known types of proximities including among others the S-proximities, LO-proximities, RI-proximities. Regular proximities of Harris, classical proximities of Efremovic as well as the basic proximities of Cech.…”

“…U will be used to denote a proximity. Basic results on grills and filters are discusssed in [16]. We begin by recalling the definition of a grill.…”