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.