For a set ‎$‎X‎$ ‎and a ‎Topological space ‎$‎(X,\tau‎)$, ‎let‎ ‎‎$‎\Gamma_{_X}(\tau)‎$ ‎be a‎ ‎graph ‎with ‎vertex ‎set ‎‎$‎\tau\setminus \{\emptyset,X\}‎$ ‎in ‎which ‎two ‎vertices ‎‎$‎A_1‎$ ‎and ‎‎$‎A_2‎$ ‎are ‎adjacent ‎just ‎when ‎‎$‎A_1\cup A_2=X‎$‎. In this paper and among some other results, we study the maximum and minimum degrees, the matching number, the chromatic number, the chromatic index, ‎the planarity, the Wiener index and the Zagreb index of ‎‎$‎\Gamma_{_X}(\tau)‎$ and we determine their exact values in general cases or in some special Topological spaces like ‎$‎T_1‎$‎‎.

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 manuscript, we introduce $LG^{c}$-fuzzy Euclidean Topological space in which $L$ denotes a completely distributive lattice with a countable subset dense in it. We use the structure of $LG$-fuzzy Topological space $(X,\ \mathfrak{T})$, which $X$ is an $L$-fuzzy subset of the crisp set $M$ and $\mathfrak{T}: L^M_X \to L $, is an $L$-gradation of openness on $X$ to define the fundamental concepts of $LG$-fuzzy analysis such as $LG$-locally compactness and $LG$-paracompactness and prove several theorems. In consequence, we show that any second countable Hausdorff $LG$-fuzzy Topological space that is $LG$-locally compact is $LG$-paracompact. Also from any given metric $\rho$ on a crisp set $M$ and $L$-fuzzy subset $X$ of it, we construct an $L$-gradation of openness $\mathfrak{T}_{\rho}$ on $X$ and obtain $LG$-fuzzy Topological metric space $(X,\mathfrak{T}_{\rho} )$. Finally, we prove an interesting theorem: Every $LG$-fuzzy Topological metric space, is $LG$-paracompact.

‎‎‎‎‎Let $H(\mathbb{D})$ be the space of all analytic functions on $\mathbb{D}$‎, ‎$u,v\in H(\mathbb{D})$ and $\varphi,\psi$ be self-map $(\varphi,\psi:\mathbb{D}\rightarrow \mathbb{D})$‎. Difference of weighted composition operator is denoted by $uC_\varphi‎ -‎vC_\psi$ and defined as follows‎ ‎\begin{align*}‎ (uC_\varphi‎ -‎vC_\psi)f(z) = u(z) f{(\varphi(z))}‎- ‎v(z) f(\psi(z))‎ ,‎\quad f\in H(\mathbb{D} )‎, ‎\quad z\in \mathbb{D}‎. ‎\end{align*}‎ ‎In this paper‎, ‎boundedness of difference of weighted composition operator from Cauchy transform into Dirichlet space will be considered and ‎an ‎equivalence condition for boundedness of such operator will be given‎.‎‎ Then the norm of composition operator between the mentioned spaces will be studied and it will be shown ‎that‎‎ $\|C_\varphi\|\geq 1$ ‎and‎ there is no composition isometry from Cauchy transform into Dirichlet ‎space‎.‎

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.