In this paper, we introduce the based and unbased n-Spanier groups and n-semilocally simply connected spaces, and investigate their relationship. We show that under some conditions, vanishing of the n-Spanier group with respect to an open cover is equivalent to the n-semilocal simple connectivity of that space, and vice versa.

No abstract

In this paper, we introduce e-right, e-left, e-semi and ε-soft topological semigroups and examine the way these are related to each other. To do so, we need to define -soft and point open soft topologies, which are defined in the third and fourth sections, respectively. Also, soft separation axioms on these soft topologies will be studied. KeywordsSoft topological semigroup • -Soft topology • Point open soft topology • T -soft space Recently, the theory has been studied and used extensively. For example, the basic aspects of soft set theory were developed in Pie and Miao (2005), Maji et al. (2003) and Chen et al. (2005), algebraic structures of soft sets were studied in Acar et al. (2010), Aktas and Cagman (2007), Jun (2008) and Sun et al. (2008), and applications of soft set theory in game theory and measure theory were given in Molodtsov et al. (2006). The study of soft topological spaces Communicated by A. Di Nola.

In this paper, we study some properties of pseudo-valuations and their induced quasi metrics. The continuity of operation of a BCK-algebra was studied with topology induced by a pseudo-valuation. Moreover, we show that product of finite number of this pseudo metric spaces is a pseudo metric space. Also, we prove that if a BCK-algebra X has a pseudo-valuation, then every quotient space of X has a pseudo metric. The completion of this spaces has been investigated in the present study. 2. Preliminaries 2.1. BCK-algebras An algebra (X, * , 0) of type (2, 0) is called a BCK-algebra if it satisfies the following axioms: for any x, y, z ∈ X, (1) ((x * y) * (x * z)) * (z * y) = 0, (2) (x * (x * y)) * y = 0,

No abstract