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In this paper, we construct a TOPOLOGY using the concept of Hoop algebras and investigate its topological properties. These include examining specific types of topological spaces, such as Hausdorff, $T_0$-spaces, and $T_1$-spaces, as well as exploring the concept of connectedness. Additionally, we analyse the relationship between closed and compact sets within this TOPOLOGY. Finally, by incorporating the binary operation $\ri$ and the defined TOPOLOGY on Hoop algebras, we introduce the notion of semi-topological algebra and demonstrate that every Hoop algebra is a right semi-topological algebra.

We define a new metric on the set of all closed linear operators between Hilbert spaces and investigate its properties. In particular, we show that the set of all closed operators with a closed range is an open subset of the set of all closed operators and the map T®T+ is an isometry in this metric. We also investigate the relationships between the TOPOLOGY induced by this metric and the gap metric.

The concept of TOPOLOGY defined on a set can be extended to the field of graph theory by defining the notion of graph topologies on graphs where we consider a collection of subgraphs of a graph G in such a way that this collection satisfies the three conditions stated similarly to that of the three axioms of point-set TOPOLOGY. This paper discusses an introduction and basic concepts to the graph TOPOLOGY. A subgraph of G is said to be open if it is in the graph TOPOLOGY TG. The paper also introduces the concept of the closed graph and the closure of graph TOPOLOGY in graph topological space using the ideas of decomposition-complement and neighborhood-complement.

The aim of this paper is to extend the truth value table of lattice-valued convergence spaces to a more general case and then to use it to introduce and study the quantale-valued fuzzy Scott TOPOLOGY in fuzzy domain theory. Let (L; ∗ ; ") be a commutative unital quantale and let ⊗ be a binary operation on L which is distributive over nonempty subsets. The quadruple (L; ∗ ; ⊗ ; ") is called a generalized GL-monoid if (L; ∗ ; ") is a commutative unital quantale and the operation ∗ is ⊗-semi-distributive. For generalized GL-monoid L as the truth value table, we systematically propose the stratified L-generalized convergence spaces based on stratified L-filters, which makes various existing lattice-valued convergence spaces as special cases. For L being a commutative unital quantale, we define a fuzzy Scott convergence structure on L-fuzzy dcpos and use it to induce a stratified L-TOPOLOGY. This is the inducing way to the definition of quantale-valued fuzzy Scott TOPOLOGY, which seems an appropriate way by some results.