A separation theorem for semicontinuous functions on preordered topological spaces

Abstract

Suppose that X is a topological space with preorder , and that −g, f are bounded upper semicontinuous functions on X such that g(x) f (y) whenever x y. We consider the question whether there exists a bounded increasing continuous function h on X such that g h f , and obtain an existence theorem that gives necessary and sufficient conditions. This result leads to an extension theorem giving conditions that allow a bounded increasing continuous function defined on an open subset of X to be extended to a function of the same type on X. The application of these results to extremally disconnected locally compact spaces is studied. Introduction.Before explaining the purpose of this paper it will be convenient to make a number of definitions. By a preorder for a set X we shall mean a reflexive, transitive relation in X, and by an order for X we shall mean an antisymmetric preorder. We shall use the symbol to denote a preorder in X, but also to denote the usual order relation for real numbers or real-valued functions. Suppose we are given a preorder for X. We say that a real-valued function f on X is increasing if f (x) f (y) whenever x, y ∈ X and x y. A subset S of X is said to be increasing if its indicator function 1 S is increasing. Decreasing functions and sets are defined analogously. Note that the complement of an increasing set is decreasing and vice versa. Given a bounded real function f on X, we denote by f ↑ the least increasing real function on X that majorizes f (see §2). Given a subset T of X we denote by i(T ) the smallest increasing subset of X that contains T and by d(T ) the smallest decreasing subset of X that contains T . If X is also a topological space, we denote by C b (X, ) the set of bounded increasing real-valued continuous functions on X, and for a bounded real function f on X we denote by f and f respectively the lower and the upper semicontinuous regularizations of f (see, for example, [4]). Finally, by a clopen set in a topological space we shall mean any subset that is open and closed, and we shall denote by E(X, ) the set of all increasing clopen subsets of X.

Continuous multi-utility representations of preorders

“…This result already has been proved by Priestley (1972). In Theorem 4.2 by Edwards (2005) a very simple proof of this result is presented. Conversely, the following proposition demonstrates that the concept of a compliant preorder leads to completely different results than the concept of a D-I-closed preorder.…”

“…This means that a Priestley (1972) and more extensively by Edwards (2002Edwards ( , 2004Edwards ( , 2005. In Bosi and Isler (2000) compliant preorders have been considered in order to prove interesting continuous utility representation theorems of the Richter-Peleg type.…”

Insertion of Continuous Real Functions on Spaces, Bispaces, Ordered Spaces and Pointfree Spaces—A Common Root

“…Given an ordered topological space (X, T, ≤) and a subset E of X, let cl↑(E) denote the smallest closed increasing subset of X containing E and let cl↓(E) denote the smallest closed decreasing subset of X containing E. The ordered space (X, T, ≤) is extremally disconnected (extremally order-disconnected in [3]) if for each increasing open subset G 1 of X the set cl↑(G 1 ) is open and for each decreasing open subset G 2 of X the set cl↓(G 2 ) is open. After realizing that (X, T, ≤) is extremally disconnected iff (X, ↑T, ↓T) is extremally disconnected iff (X, ↓T, ↑T) is extremally disconnected, one can obtain immediately the following theorem as a particular case of our Theorem 6.4 above (cf.…”

“…After realizing that (X, T, ≤) is extremally disconnected iff (X, ↑T, ↓T) is extremally disconnected iff (X, ↓T, ↑T) is extremally disconnected, one can obtain immediately the following theorem as a particular case of our Theorem 6.4 above (cf. [3]):…”