In this paper we give some characterizations of TOPOLOGICAL extreme amenability. Also we answer a question raised by Ling [5]. In particular we prove that if T is a Borel subset of a locally compact semigroup S such that M(S)* has a multiplicative TOPOLOGICAL left invariant mean then T is TOPOLOGICAL left lumpy if and only if there is a multiplicative TOPOLOGICAL left invariant mean M on M(S)* such that M(XT)=1, where XT is the characteristic functional of T. Consequently if T is a TOPOLOGICAL left lumpy locally compact Borel subsemigroup of a locally compact semigroup S, then T is extremely TOPOLOGICAL left amenable if and only if S is.

In this paper, we study the separtion axioms T 0, T 1, T 2 and T 5/2 on TOPOLOGICAL and semiTOPOLOGICAL residuated lattices and we show that they are equivalent on TOPOLOGICAL residuated lattices. Then we prove that for every infinite cardinal number a, there exists at least one nontrivial Hausdor TOPOLOGICAL residuated lattice of cardinality a. In the follows, we obtain some conditions on (semi) TOPOLOGICAL residuated lattices under which this spaces will convert into regular and normal spaces.Finally by using of regularity and normality, we convert (semi) TOPOLOGICAL residuated lattices into metrizable TOPOLOGICAL residuated lattices.

In this paper we introduce the concept of TOPOLOGICAL number for locally convex TOPOLOGICAL spaces and prove some of its properties. It gives some criterions to study locally convex TOPOLOGICAL spaces in a discrete approach.

Hyperdiamonds are covalently bonded fullerenes in crystalline forms, more or less related to diamond, and having a significant amount of sp3 carbon atoms. Design of several hypothetical crystal networks was performed by using our original software programs CVNET and NANO-STUDIO. The topology of the networks is described in terms of the net parameters and several counting polynomials, calculated by NANO-STUDIO, OMEGA and PI software programs.

In addition to exploring constructions and properties of limits and colimits in categories of TOPOLOGICAL algebras, we study special subcategories of TOPOLOGICAL algebras and their properties. In particular, under certain conditions, reflective subcategories when paired with TOPOLOGICAL structures give rise to reflective subcategories and epireflective subcategories give rise to epireflective subcategories.

In this paper, we have defined and studied a generalized form of TOPOLOGICAL vector spaces called s-TOPOLOGICAL vector spaces. s-TOPOLOGICAL vector spaces are defined by using semi-open sets and semi-continuity in the sense of Levine. Along with other results, it is proved that every s-TOPOLOGICAL vector space is generalized homogeneous space. Every open subspace of an s-TOPOLOGICAL vector space is an s-TOPOLOGICAL vector space. A homomorphism between s-TOPOLOGICAL vector spaces is semi-continuous if it is s-continuous at the identity.

One of the most important parts of TOPOLOGICAL structures is metrizability, which creates the conditions for a metric such as d to be found so that the topology obtained from this metric is the same as the original topology. TOPOLOGICAL algebraic structures that are closely related to ℝ,and its resulting spaces. For example, in TOPOLOGICAL groups, as a TOPOLOGICAL algebraic structure, there are very close properties with metric and soft spaces, such as being completely regular, being Hausdorff, …, . . In terms of metrizability, there is an astonishing case called the Birkhoff-Kakutani case, which states that a TOPOLOGICAL group is metricable if and only if it is the first-countable. In this manuscript, we make TOPOLOGICAL polygroup metrizable. We do this by definition of the prenorm on TOPOLOGICAL polygoups, and finally we prove the fundamental theorem of metrizability on TOPOLOGICAL polygroups Like TOPOLOGICAL groups, it is the first-countable of TOPOLOGICAL polygroup.

The main purpose of this paper is to introduce a concept of L- fuzzifying TOPOLOGICAL groups (here L is a completely distributive lattice) and discuss some of their basic properties and the structures. We prove that its corresponding L-fuzzifying neighborhood structure is translation invariant. A characterization of such TOPOLOGICAL groups in terms of the corresponding L- fuzzifying neighborhood structure of the unit is given. It is shown that the category of L-fuzzifying TOPOLOGICAL groups L-FYTPG is TOPOLOGICAL over the category of groups GRP with respect to the forgetful functor. As an application, the conclusion that the product of L-fuzzifying TOPOLOGICAL groups is also an L-fuzzifying TOPOLOGICAL group is proved. Finally, it is proved the forgetful functor preserves the product.