This paper establishes two new connections between the familiar function ring functor ${mathfrak R}$ on the category ${bf CRFrm}$ of completely regular frames and the category {bf CR}${mathbf sigma}${bf Frm} of completely regular $sigma$-frames as well as their counterparts for the analogous functor ${mathfrak Z}$ on the category {bf ODFrm} of 0-dimensional frames, given by the integer-valued f...

In point set topology—or, more precisely, in the part of it once called "theory of abstract spaces"—there are two principal methods of investigation. The first one refers to the topological space alone(2) ; we shall call it the internal method : for example the separation axioms Po, Pi, T2 [A-H], the notions of regularity, normality, compactness, are expressed in terms of the topological space ...

a necessary and sufficient condition in terms of lower cut sets are given for the insertion of a $gamma-$continuous function between two comparable real-valued functions.

The set of Hausdorff continuous functions is the largest set of interval valued functions to which the ring structure of the set of continuous real functions can be extended. The paper deals with the automation of the algebraic operations for Hausdorff continuous functions using an ultraarithmetical approach.

Let A be a local ring with 2 invertible. It is known that the localization of the cohomology ring H∗ ét(A,Z/2) with respect to the class (−1) ∈ H1 ét(A,Z/2) is isomorphic to the ring C(sperA,Z/2) of continuous Z/2-valued functions on the real spectrum of A. Let In(A) denote the powers of the fundamental ideal in the Witt ring of symmetric bilinear forms over A. The starting point of this articl...

Let $L$ be a completely regular frame and $mathcal{R}L$ be the ‎ring of continuous real-valued functions on $L$‎. ‎We show that the‎ ‎lattice $Zid(mathcal{R}L)$ of $z$-ideals of $mathcal{R}L$ is a‎ ‎normal coherent Yosida frame‎, ‎which extends the corresponding $C(X)$‎ ‎result of Mart'{i}nez and Zenk‎. ‎This we do by exhibiting‎ ‎$Zid(mathcal{R}L)$ as a quotient of $Rad(mathcal{R}L)$‎, ‎the‎ ‎...

Let X be a completely regular space and let A(X) be a ring of continuous real-valued functions onX which is closed under local bounded inversion. We show that the structure space of A(X) is homeomorphic to a quotient of the Stone–Čech compactification of X. We use this result to show that any realcompactification of X is homeomorphic to a subspace of the structure space of some ring of continuo...