If U and V are topologies on an abstract set X, then the triple (X, U, V) is a bitopological space. Using the theorem of Priestley on the representation of distributive lattices, results of Dilworth concerning the normal completion of the lattice of bounded, continuous, realvalued functions on a topological space are extended to include the lattice of bounded, semi-continuous, real-valued funct...

Extrinsic and intrinsic characterizations are given for the class DSC(K) of differences of semi-continuous functions on a Polish space K, and also decomposition characterizations of DSC(K) and the class PS(K) of pointwise stabilizing functions on K are obtained in terms of behavior restricted to ambiguous sets. The main, extrinsic characterization is given in terms of behavior restricted to som...

In the main theorem of this paper we show that any involution on the class of lower semi-continuous convex functions which is order-reversing, must be, up to linear terms, the well known Legendre transform.

In this paper, we introduce new class of functions called τ∗-Generalized Semi Continuous functions in topological spaces and study some of its properties and its relationship with some existing functions. Mathematics Subject Classification: 54A05

Starting with the intuitive concept o r nearness" as a binar,, relation, semi-proximity spaces (sp-spaces) are defined. The restrictions on semi-proximity spaces are weaker than on topological proximity spaces. Thus, semi-proximity spaces generalize classical topological spaces. Moreover, it is possible to describe all digital pictures used in computer vision and computer graphics as non-trivia...

RECENTLY G. C. Young* and A. Denjoyf have communicated theorems—those in Denjoy's memoir are of an especially comprehensive character—dealing, in particular, with point sets where the four derivatives of a given continuous function are identical. It is the purpose of this note to treat an analogous problem that arises when "derivative" is replaced by "saltus."$ However, instead of confining our...

We prove that an n-valued function that is lower semi-continuous is also upper semi-continuous and therefore continuous, that is, an n-valued map. Several conditions are shown to be equivalent to continuity for n-valued functions, in particular the continuity of a corresponding singlevalued function to a configuration space. Consequently, splitting is related to braid groups to prove a Splittin...

In this paper, we will focus on three types of functions in a generalized topological space, namely; lower and upper semi-continuous functions, cliquish functions. We give some results for nowhere dense sets second category sets. Further, discuss the nature metric spaces provide characterization theorem terms