Mathematics

We construct a metrizable semitopological semilattice X whose partial order P={(x,y)∈X×X:xy=x} is a non-closed dense subset of X×X . As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.

In accordance with M 3 -structures in paper [4], we construct a stratifiable space which is not M 1 -spaces.

In this paper we derive coincidence and common fixed point results under order homotopies of families of mappings in preordered b -metric spaces.

We give a new proof of the Katětov-Tong theorem. Our strategy is to first prove the theorem for compact Hausdorff spaces, and then extend it to all normal spaces. The key ingredient is how the ring of bounded continuous real-valued functions embeds in the ring of all bounded real-valued functions. In the compact case this embedding can be described by an appropriate statement, which we prove implies both the Katětov-Tong theorem and a version of the Stone-Weierstrass theorem. We then extend the Katětov-Tong theorem to all normal spaces by showing how to extend upper and lower semicontinuous real-valued functions to the Stone-\v Cech compactification so that the less than or equal relation between the functions is preserved.

In this article, we present a new characterization of the completeness of a partial metric space--which we call \textit{orbital characterization}-- using fixed point results.

In this paper, we introduce a new method for compactification of a topological space by order topology and through ordinal numbers. The idea behind our approach originates from the definition of a limit point, and then we try to find an intuition for this concept. Finally, we utilise the Homotopy concept for separation Axiom

In a paper by Bella, Tokgös and Zdomskyy it is asked whether there exists a Tychonoff space X such that the remainder of C p (X) in some compactification is Menger but not σ -compact. In this paper we prove that it is consistent that such space exists and in particular its existence follows from the existence of a Menger ultrafilter.

The note contains a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is presented a semiregular semitopological group G which is not T 3 . We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed examples of quasiregular T 1 compact and T 2 sequentially compact quasitopological groups, which are not paratopological groups. Also we prove that a semitopological group (G,τ) is a topological group provided there exists a Hausdorff topology σ⊃τ on G such that (G,σ) is a precompact topological group and (G,τ) is weakly semiregular or (G,σ) is a feebly compact paratopological group and (G,τ) is T 3 .

The result of Boyce and Huneke gives rise to a 1-dimensional continuum, which is the intersection of a descending family of disks, that admits two commuting homeomorphisms without a common fixed point.

In this paper we shall give a short proof of the result originally obtained by Ashutosh Kumar that for each A⊂R there exists B⊂A full in A such that no distance between two distinct points from B is rational. We will construct a Bernstein subset of R which also avoids rational distances. We will show some cases in which the former result may be extended to subsets of R 2 , i. e. it remains true for measurable subsets of the plane and if non(N)=cof(N) then for a given set of positive outer measure we may find its full subset which is a partial bijection and avoids rational distances.