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.

Many topological objectives and constraints cannot easily be assessed by analytical formulations. This paper introduces a number of GRAPH theoretical operators, as suitable combinatorial tools, for discrete TOPOLOGY assessment. Using such an approach, the load paths from their exertion points to the support joints can be guided topologically during the optimization process. Eleven variants of the proposed method are developed for the inclusion of various constraints and/or objectives. The presented algorithms are then applied to the optimal bracing layout of multi-story frames under lateral loadings for minimal weight or static compliance. Benchmark examples from literature are treated to validate the efficiency and to compare the capability of the proposed algorithms. The bracing patterns obtained from optimization are GRAPH theoretically categorized.

Let G = (V, E) be a locally finite GRAPH, i.e. a GRAPH in which every vertex has finitely many adjacent vertices. In this paper, we associate a TOPOLOGY to G, called GRAPHic TOPOLOGY of G and we show that it is an Alexandro TOPOLOGY, i.e. a TOPOLOGY in which intersection of each family of open sets is open. Then we investigate some properties of this TOPOLOGY. Our motivation is to give an elementary step toward investigation of some properties of locally finite GRAPHs by their corresponding TOPOLOGY which we introduce in this paper.

An MI-group is an algebraic structure based on a generalization of the con-cept of a monoid that satis es the cancellation laws and is endowed with an invertible anti-automorphism representing inversion. In this paper, a TOPOLOGY is de ned on an MI-group G under which G is a topological MI-group. Then we will identify open, discrete and compact MI-subgroups. The connected components of the elements of G and connected MI-groups are also identi ed. Some features of the maximal MI-subgroups and ideals of a topological MI-group are investigated as well. Finally, some theorems about automatic continuity will be introduced.