We shall find the following necessary and sufficient conditions: I. the space is metrizable (cf. Nagata [l], Smirnof [2]), II. the space is strongly O-dimensional. Property II means that any two closed disjoint sets in the space can be separated (by the empty set). We shall prove furthermore that the conditions I and II are equivalent to the following topological properties: the space is a Haus...

The goal of this paper is to investigate the certain properties bipolar fuzzy supra preopen sets (BF set). We introduce and study concepts pre-interior pre-interior) pre-closure pre-closure) operators. Furthermore, we define new type continuous mapping based on BF sets.

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhood, the sets of colors that appear in the closed neighborhood of u and v are distinct. Let χlid(G) be the minimum number of colors used in a locally identifying vertex-coloring of G. In ...

A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Cla...

Inspired by questions of convergence in continued fraction theory, Erdős, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Möbius maps acting on the Riemann sphere, S. By identifying S with the boundary of three-dimensional hyperbolic space, H, we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of H. Usin...

Topological measures and quasi-linear functionals generalize linear functionals. We define study deficient topological on locally compact spaces. A measure a space is set function open closed sets which finitely additive sets, inner regular outer sets. Deficient measures. First we investigate positive, negative, total variation of signed that only assumed to be These variations turn out Then ex...

The main aim of this work is to introduce and study the notions generalized regularity, normality, symmetric in fuzzy soft topological spaces via closed sets. Investigating some their basic properties. Many related theorems relations these are presented. Moreover, hereditary property preservation discussed.