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Abstract  

A number of generalized metric spaces have been defined or characterized in terms of g-functions. Symmetric g-functions are discussed by C. Good, D. Jennings and A. M. Mohamad. In this paper, some questions about symmetric g-functions are answered, particularly it is shown that every sym-wg-space is expandable.

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We show that the cardinality of power homogeneous T 5 compacta X is bounded by 2 c ( X ) . This answers a question of J. van Mill, who proved this bound for homogeneous T 5 compacta. We further extend some results of I. Juhász, P. Nyikos and Z. Szentmiklóssy and as a corollary we prove that consistently every power homogeneous T 5 compactum is first countable. This improves a theorem of R. de la Vega who proved this consistency result for homogeneous T 5 compacta.

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Abstract  

The concept of normality is defined for generalized topologies in the sense of [1], a few properties of normal spaces are proved, and their characterization with the help of a suitable form of Urysohn’s lemma is discussed.

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Abstract

A new kind of sets called generalized μ-closed (briefly g μ-closed) sets are introduced and studied in a topological space by using the concept of generalized open sets introduced by Á. Császár. The class of all g μ-closed sets is strictly larger than the class of all μ-closed sets. Furthermore, g-closed sets (in the sense of N. Levine [17]) is a special type of g μ-closed sets in a topological space. Some of their properties are investigated. Finally, some characterizations of μ g-regular and μ g-normal spaces have been given.

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Abstract

We introduce new types of sets called -sets and -sets and study some of their fundamental properties. We then investigate the topologies obtained from these sets.

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Abstract  

Insertion of lattice-valued functions in a monotone manner is investigated. For L a ⊲-separable completely distributive lattice (i.e. L admits a countable base which is free of supercompact elements), a monotone version of the Katětov-Tong insertion theorem for L-valued functions is established. We also provide a monotone lattice-valued version of Urysohn’s lemma. Both results yield new characterizations of monotonically normal spaces. Moreover, extension of lattice-valued functions under additional assumptions is shown to characterize also monotone normality.

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Abstract  

A sufficient condition for the strict insertion of a continuous function between two comparable upper and lower semicontinuous functions on a normal space is given. Among immediate corollaries are the classical insertion theorems of Michael and Dowker. Our insertion lemma also provides purely topological proofs of some standard results on closed subsets of normal spaces which normally depend upon uniform convergence of series of continuous functions. We also establish a Tietze-type extension theorem characterizing closed G δ-sets in a normal space.

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Acta Mathematica Hungarica
Authors: J. Gutierrez Garcia, I. Mardones-Pérez, and M. De Prada Vicente

Abstract  

Monotone normality is usually defined in the class of T 1 spaces. In this paper new characterizations of monotone normality, free of T 1 axiom, are provided and it is shown that in this context it is not a hereditary property. Also, a Tietze-type extension theorem for lattice-valued functions for this class of spaces is given.

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