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In 1964 Frink [8] generalized Wallman's method [17] of compactification and asked the question: “Is every Hausdorff compactification of a Tychonoff space a Wallman compactification?”. This problem, which is as yet unsolved, has led to the discovery of a number of necessary and/or sufficient conditions for a Hausdorff compactification to be Wallman; (see Alò and Shapiro [2], [3], Banasĉhewski [6], Njåstad [13] and Steiner [15]). Recently Alò and Shapiro [4], [5] have generalized the Wallman procedure to discuss what they call Z*-realcompact spaces and Z*-realcompactification n(Z) corresponding to a countably productive (c.p.) normal base Z on X. Some further work has been done by Steiner and Steiner [14].

Proximity, contiguity and nearness structures are here studied from a unified point of view. In the discussion the role that grills can play in the theory is emphasized. Nearness structures were recently introduced by Herrlich and Naimpally. Thron pointed out the importance of grills in proximity theory. Nearness structures v v are then used to generate proximity extensions ( ϕ , ( X v , Π v ) ) (\phi ,({X^v},{\Pi ^v})) of a given LO-proximity space ( X , Π ) (X,\Pi ) , where Π v = Π {\Pi _v} = \Pi . Finally, the relation of the extensions ( ϕ , ( X v , Π v ) ) (\phi ,({X^v},{\Pi ^v})) to arbitrary extensions ( i , ( Y , Π ∗ ) ) (i,(Y,{\Pi ^ \ast })) is investigated.

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