We show that the cardinality of power homogeneous
is bounded by 2
. This answers a question of J. van Mill, who proved this bound for homogeneous
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
compactum is first countable. This improves a theorem of R. de la Vega who proved this consistency result for homogeneous
The concept of normality is defined for generalized topologies in the sense of , a few properties of normal spaces are
proved, and their characterization with the help of a suitable form of Urysohn’s lemma is discussed.
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 ) is a special type of g μ-closed sets in a topological space. Some of their properties are investigated. Finally, some characterizations of μg-regular and μg-normal spaces have been given.
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 Gδ-sets in a normal space.
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.
Authors:J. Gutierrez Garcia, I. Mardones-Pérez, and M. De Prada Vicente
Monotone normality is usually defined in the class of T1 spaces. In this paper new characterizations of monotone normality, free of T1 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.