Following the introduction of separability in frames ([2]) we investigate further properties of this notion and establish some consequences of the Urysohn metrization theorem for frames that are frame counterparts of corresponding results in spaces. In particular we also show that regular subframes of compact metrizable frames are metrizable.
We prove a theorem that generalizes in a way both Michael's Selection Theorem and Dugundji's Simultaneous Extension Theorem. We use it to prove that if K is an uncountable compact metric space and X a Banach space, then C(K,X) is isomorphic to C(𝓒,X) where 𝓒 denotes the Cantor set. For X = ℝ, this gives the well known Milyutin Theorem.
In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number , a topological group G such that is countably compact for all cardinals γ < α, but is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from . However, the question has remained...
We prove that a map between two realcompact spaces is skeletal if and only if it is homeomorphic to the limit map of a skeletal morphism between ω-spectra with surjective limit projections.
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.
It is proved that the class of separable Rosenthal compacta on the Cantor set having a uniformly bounded dense sequence of continuous functions is strongly bounded.
The universality problem focuses on finding universal spaces in classes of topological spaces. Moreover, in “Universal spaces and mappings” by S. D. Iliadis (2005), an important method of constructing such universal elements in classes of spaces is introduced and explained in details. Simultaneously, in “A topological dimension greater than or equal to the classical covering dimension” by D. N. Georgiou, A. C. Megaritis and F. Sereti (2017), new topological dimension is introduced and studied, which...