Normal spaces are characterized in terms of an insertion type theorem, which implies the Katětov-Tong theorem. The proof actually provides a simple necessary and sufficient condition for the insertion of an ordered pair of lower and upper semicontinuous functions between two comparable real-valued functions. As a consequence of the latter, we obtain a characterization of completely normal spaces by real-valued functions.
Insertion and extension theorems are presented for lattice-valued functions where the lattice is an appropriately based completely distributive lattice endowed with its interval topology.
A family of subsets of a set is called a -topology if it is closed under arbitrary countable unions and arbitrary finite intersections. A -topology is perfect if any its member (open set) is a countable union of complements of open sets. In this paper perfect -topologies are characterized in terms of inserting lower and upper measurable functions. This improves upon and extends a similar result concerning perfect topologies. Combining this characterization with a -topological version of Katětov-Tong...
Pointfree formulas for three kinds of separating points for closed sets by maps are given. These formulas allow controlling the amount of factors of the target product space so that it does not exceed the weight of the embeddable space. In literature, the question of how many factors of the target product are needed for the embedding has only been considered for specific spaces. Our approach is algebraic in character and can thus be viewed as a contribution to Kuratowski's topological calculus.
F. van Gool [Comment. Math. Univ. Carolin. (1992), 505–523] has introduced the concept of lower semicontinuity for functions with values in a quasi-uniform space . This note provides a purely topological view at the basic ideas of van Gool. The lower semicontinuity of van Gool appears to be just the continuity with respect to the topology generated by the quasi-uniformity , so that many of his preparatory results become consequences of standard topological facts. In particular, when the order...
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