In this paper, a concept of nearness convergence is introduced which contains the proximal convergence of Leader as a special case. It is proved that uniform convergence and this nearness convergence are equivalent on totally bounded uniform nearness spaces. One of the features of this convergence is that it lies between uniform convergence and pointwise convergence, and implies uniform convergence on compacta. Some other weaker notions of nearness convergence which are sufficient to preserve nearness maps are also discussed.

Recent work by Krystock, Porter, and Vermeer has emphasized the importance of the concepts of Katětov spaces and H-sets in the theory of H-closed spaces. These properties are closely related to being the θ-closure of some set and being the adherence of an open filter. This relationship is developed by establishing, among other facts, that an H-closed space in which every closed set is the θ-closure of some set is compact and the θ-closure of a subset of an H-closed space is Katětov and characterizing the open filter adhérences of a space as precisely those sets which are the image of a closed set of the absolute of the space. Also, examples are given of a countable, scattered space which is not Katětov and an H-closed space with an H-closed subspace which is not the θ-closure of any subset of the given space.

It is a question of Arhangel'skiĭ [1] (Problem 2) whether the identity ψ(G) = X(G) holds for every minimal Hausdorff topological group G = 〈G,u〉). (Here, as usual, ψ(G), the pseudocharacter of G, is the least cardinal number K for which there is such that and and x(G), the character of G,is the least cardinality of a local base at e for (〈G,u〉.) That 〈G, u〉 is minimal means that, if v is a Hausdorff topological group topology for G and v ⊂ u, then v = u.

In this paper, we give some conditions on G sufficient to ensure a positive response to Arhangel'skiï's question, and we offer an example which responds negatively to a question on minimal groups posed some years ago (cf. [6] (p. 107) and [4] (p. 259)).

New characterizations of weakly-continuous and θ-continuous functions are presented, and θ-continuity is applied to characterize H(i) spaces; a recent characterization of closed graph functions is utilized to characterize H-closed spaces. Noiri has shown that a function λ which is almost-continuous in the sense of Husain is weakly-continuous if cl(λ−1(W)) ⊂ λ−1(cl(W)) for all open W. It is established here that almost-continuity is superfluous in this statement.

In this paper the notion of inverse cluster set, which was recently introduced and studied for functions by T. R. Hamlett and P. E. Long (Proc. Amer. Math. Soc, 53 (1975), 470-476), is extended to and investigated for multifunctions. We generalize the notion of inverse cluster set, extend to multifunctions and generalize some known results for inverse cluster sets of functions and offer some new results. In the latter sections, compactness generalizations are characterized in terms of inverse cluster sets and some results on connected and conectivity functions are extended to multifunctions.

This paper contains several characterizations of regular-closed and minimal regular topological spaces along with some related properties.