Continuity and Separation for Pointwise-symmetric Isotonic Closure Functions

In this paper, we show that a pointwise-symmetric isotonic closure function is uniquely determined by the pairs of sets it separates. We then show that when the closure function of the domain is isotonic and the closure function of the codomain is isotonic and pointwise-symmetric, functions which separate only those pairs of sets which are already separated are continuous, generalizing a result in [1] which is, in turn, a generalization of a result in [2]. A generalized closure space is a pair (X, cl) consisting of a set X and a closure function cl, a function from the power set of X to itself. The closure of a subset A of X, denoted cl(A), is the image of A under cl. The exterior of A is ext(A) = X − cl(A), and the interior of A is int(A) = X − cl(X − A). We say that A is closed if A = cl(A), A is open if A = int(A), and N is a neighborhood of x if x ∈ int(N). In this paper, we use terminology for generalized closure spaces found in [3], namely, we say that a closure function cl defined on X is (1) grounded, if cl(∅) = ∅. (2) isotonic, if cl(A) ⊆ cl(B) whenever A ⊆ B. (3) enlarging, if A ⊆ cl(A), for each subset A of X. (4) idempotent, if cl(A) = cl(cl(A)), for each subset A of X. (5) sub-linear, if cl(A ∪ B) ⊆ cl(A) ∪ cl(B), for all A, B ⊆ X. Additionally, we define a kind of separation useful for the task at hand: Definition 1. Subsets A and B of X are said to be closure-separated in a generalized closure space (X, cl) (or simply, cl-separated) if A ∩ cl(B) = ∅ and cl(A) ∩ B = ∅, or, equivalently, if A ⊆ ext(B) and B ⊆ ext(A).