The Sorgenfrey line has a locally pathwise connected connectification

We answer a question of Alas, Tkacenko, Tkachuk and Wilson by constructing a connected locally pathwise connected Hausdorff space in which the Sorgenfrey line can be densely embedded.

     

Constancy of upper-continuous two-valued mappings into the Sorgenfrey line

By using the Sierpiński continuum theorem, we prove that every upper-continuous two-valued mapping of a linearly connected space (or even a c-connected space, i.e., a space in which any two points can be connected by a continuum) into the Sorgenfrey line is necessarily constant. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1034–1039, August, 2007.     

A non-metrizable space whose countable power is -metrizable

We answer a question of A.V. Arhangel'skii by finding a non-metrizable space such that is the countable union of metrizable spaces.

     

Hartman-Mycielski functor of non-metrizable compacta

We investigate certain topological properties of the normal functor H, introduced by the first author, which is a certain functorial compactification of the Hartman-Mycielski construction HM. We prove that H is always open and we also find the condition when H X is an absolute retract, homeomorphic to the Tychonov cube.     

Variational principles in non-metrizable spaces

We study generic variational principles in optimization when the underlying topological space X is not necessarily metrizable. It turns out that, to ensure the validity of such a principle, instead of having a complete metric which generates the topology in the space X (which is the case of most variational principles), it is enough that we dispose of a complete metric on X which is stronger than the topology in X and fragments the space X.     

Perfectly normal non-metrizable non-Archimedean spaces are generalized Souslin lines

In this paper we prove the equivalence between the existence of perfectly normal, non-metrizable, non-archimedean spaces and the existence of ``generalized Souslin lines", i.e., linearly ordered spaces in which every collection of disjoint open intervals is -discrete, but which do not have a -discrete dense set. The key ingredient is the observation that every first countable linearly ordered space has a dense non-archimedean subspace.

     

Linear homeomorphisms of spaces of continuous functions on long Sorgenfrey lines

We carry out a linear homeomorphic classification of the spaces of continuous functions on the long Sorgenfrey lines S_α, where a is an arbitrary ordinal. The spaces of continuous functions are endowed with the topology of pointwise convergence and denoted by C_p(S_α).     

On a Homeomorphism between the Sorgenfrey Line S and Its Modification S P

Mathematical Notes - A topological space S P , which is a modification of the Sorgenfrey line S, is considered. It is defined as follows: if x ∈ P ? S, then a base of neighborhoods of x...     

On the ω-problem for some types of non-metrizable spaces

The ω-problem on a topological space X consists in finding out whether there exists a function whose oscillation is equal to a given upper semi-continuous (USC) function f:X→[0,∞] vanishing at isolated points of X. If such F exists, we call it an ω-primitive for f. Unlike the case of metrizable spaces, an ω-primitive need not exist if X is not metrizable. We study the ω-problem for f taking the value ∞ in the case of ordinal space, products of regular “constancy” spaces and the wedge sums of such spaces. Some open problems are formulated.     

The line digraph of a regular and pancircular digraph is also regular and pancircular

This paper introduces thepancircularity property on digraphs: a digraphD is said to bepancircular if it contains circuits of every lengthL for all 1 L # E(D). We discuss preservation of pancircularity under the line-digraph operation, and prove the theorem stated in the title. As a corollary, all DeBruijn graphs are proved to be pancyclic and pancircular.     

Every 3-connected essentially 10-connected line graph is Hamilton-connected

Thomassen conjectured that every 4-connected line graph is Hamiltonian. Lai et al. (in 2006) [5] considered whether the high essential connectivity of the 3-connected line graphs can guarantee the existence of the Hamiltonian cycle in graphs and they showed that every 3-connected, essentially 11-connected line graph is Hamiltonian. In this note, we show that every 3-connected, essentially 10-connected line graph is Hamiltonian-connected. The result strengthens the known result mentioned above.     

The line analog of Ramsey numbers

For positive integersr ands, f′(r, s) is defined as the smallest positive integerp such that every connected (ordinary) graph of orderp contains eitherr mutually adjacent lines ors mutually disjoint lines. It is found thatf’(r,s) =(r−1) (s−1)+2 unlessr=2 and s ≠ 1, in which casef′(2,s)=3. Definitions not given here can be found in [7, 8].