Ordered Cauchy spaces.
Ordered compactification of totally ordered spaces.
Ordered compactifications and families of maps.
Ordered spaces and quasi-uniformities on spaces of continuous order-preserving functions.
In this paper we introduce and investigate the notions of point open order topology, compact open order topology, the order topology of quasi-uniform pointwise convergence and the order topology of quasi-uniform convergence on compacta. We consider the functorial correspondence between function spaces in the categories of topological spaces, bitopological spaces and ordered topological spaces. We obtain extensions to the topological ordered case of classical topological results on function spaces....
Ordered spaces with special bases
We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space has a weakly uniform base if and only if it is quasi-developable and has a -diagonal, that a linearly ordered space has a point-countable base if and only if it is first-countable and has an ω-in-ω base, and that metrizability in a generalized ordered...
Order-like structure of monotonically normal spaces
For a compact monotonically normal space X we prove: (1) has a dense set of points with a well-ordered neighborhood base (and so is co-absolute with a compact orderable space); (2) each point of has a well-ordered neighborhood -base (answering a question of Arhangel’skii); (3) is hereditarily paracompact iff has countable tightness. In the process we introduce weak-tightness, a notion key to the results above and yielding some cardinal function results on monotonically normal...
Order-theoretical connectivity.
Ordinal invariants and epimorphisms in some categories of weak Hausdorff spaces
Ordinal invariants in topology II. Sequential order of compactifications
Ordinal remainders of classical ψ-spaces
Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain of infinite subsets of ω, there exists , an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ∖ψ of the associated ψ-space, ψ = ψ(ω,ℳ ), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < ⁺, where is the tower number, there exists a mod-finite ascending chain , hence a ψ-space with Stone-Čech remainder...
Oб одной задаче Э. Чеха
-regular Cauchy completions.
-sequential like properties in function spaces
We introduce the properties of a space to be strictly or strictly , where , and we analyze them and other generalizations of -sequentiality () in Function Spaces, such as Kombarov’s weakly and strongly -sequentiality, and Kocinac’s and -properties. We characterize these in in terms of cover-properties in ; and we prove that weak -sequentiality is equivalent to -property, where and , in the class of spaces which are -compact for every ; and that is a -space iff satisfies...
-topological and -regular: Dual notions in convergence theory.
-topological Cauchy completions.
Pairwise monotonically normal spaces.
Pairwise monotonically normal spaces
We introduce and study the notion of pairwise monotonically normal space as a bitopological extension of the monotonically normal spaces of Heath, Lutzer and Zenor. In particular, we characterize those spaces by using a mixed condition of insertion and extension of real-valued functions. This result generalizes, at the same time improves, a well-known theorem of Heath, Lutzer and Zenor. We also obtain some solutions to the quasi-metrization problem in terms of the pairwise monotone normality.
Pairwise s-normal spaces
Pairwise weakly Hausdorff spaces
In this paper, we introduce and investigate the notion of weakly Hausdorffness in bitopological spaces by using the convergent of nets. Several characterizations of this notion are given. Some relationships between these spaces and other spaces satisfying some separation axioms are studied.
Pairwise weakly regular-Lindelöf spaces.