Intrinsic characterizations of C-realcompact spaces

<pre>c-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41, no. 8 (2018), 1135-1167. We offer a characterization of these spaces X via c-stable family of closed sets in X by showing that X is c-realcompact if and only if each c-stable family of closed sets in X with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a c-realcompact space. We show that each topological space can be extended as a dense subspace to a c-realcompact space with some desired extension properties. An allied class of spaces viz CP-compact spaces akin to that of c-realcompact spaces are introduced. The paper ends after examining how far a known class of c-realcompact spaces could be realized as CP-compact for appropriately chosen ideal P of closed sets in X.</pre>

UNIFORM EXHAUSTIVITY OF A FAMILY OF REGULAR SET FUNCTIONS ON TOPOLOGICAL SPACES

Conditions for the uniform exhaustivity of a family of regular set functions defined on an algebra £ of subsets of a cr-topological space and taking values in arbitrary topological space are found.

EXTENSION OF CONTINUOUS MAPPINGS AND H1-RETRACTS

We prove that any continuous mapping f:E→Y on a completely metrizable subspace E of a perfect paracompact space X can be extended to a Lebesgue class one mapping g:X→Y (that is, for every open set V in Y the preimage g−1(V ) is an Fσ-set in X) with values in an arbitrary topological space Y.

Elementary topology and universal computation

This paper attempts to define a general framework for computability on an arbitrary topological space X . The elements of X are taken as primitives in this approach—also for the coding of functions — and, except when X = N, the natural numbers are not used directly.

Singular Integrals are Perron Integrals of a Certain Type

In [7] a Perron-like integral was denned in an arbitrary topological space and many of its basic properties were established. In this paper we shall show (the theorem in § 2) that in a suitable setting the integral from [7] includes a class of so-called singular integrals, i.e., generalized forms of the Cauchy principal value of an integral. Thus, the powerful machinery of Perron integration, e.g., the monotone and dominant convergence theorems, can be automatically applied to these singular integrals.