We prove that the space of causal curves between compact subsets of a separable globally hyperbolic poset is itself compact in the Vietoris topology. Although this result implies the usual result in general relativity, its proof does not require the use of geometry or differentiable structure.

The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topology, and the Polish space of compact subsets of the closed unit disc of the complex plane when equipped with the Hausdorff topology. Then, it is shown that the unit spectrum functi...

There is a compact solenoidal group S that is group isomorphic to the additive real numbers R. The existence of 5 leads to a study of compact real groups; that is, compact groups which are group isomorphic to R. The compact real groups are characterized as products of 5. Various conditions on a topological group G are given which are equivalent to G being compact real. It is shown that any comp...

Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive C0-semigroups under topology uniform weak operator convergence on compact subsets R+. Eisner and Serény raised in [6] [3] open problem: Does this constitute a Baire space? Observing that subspace unitary semigroups is completely metrisable appealing to known density resul...

Let X be a separable complete metric space. We characterize completely the innnitesimal generators of semigroups of linear transformations in C b (X), the bounded real-valued continuous functions on X, that are induced by strongly continuous semigroups of continuous transformations in X. In order to do this, C b (X) is equipped with a locally convex topology known as the strict topology. Introd...

This study proposes a controller design method based on block sparse optimization for dynamical network systems. The objective of the is to stabilize systems with given convergence rate. minimizes number controlled nodes. unique in that structure constrained by topology system. Additionally, proposed problem separable terms distributed over networks. applicable pinning control consensus and opt...

We proposed a framework for the topological classification of non-Hermitian systems. Different from previous $K$-theoretical approaches, our approach is homotopy classification, which enables us to see more invariants. Specifically, we considered systems with separable band structures. found that whole set decomposed into several sectors based on braiding energy levels and characterized by some...

Cardinal invariants related to sequential separability of generalized Cantor cubes 2κ, introduced by M. Matveev, are studied here. In particular, it is shown that the following assertions are relatively consistent with ZFC: (1) 2ω1 is sequentially separable, yet there is a countable dense subset of 2ω1 containing no non-trivial convergent subsequence, (2) 2ω1 is not sequentially separable, yet ...

We investigate various kinds of bases in infinite dimensional Banach spaces. In particular, we consider the complexity of Hamel bases in separable and non-separable Banach spaces and show that in a separable Banach space a Hamel basis cannot be analytic, whereas there are non-separable Hilbert spaces which have a discrete and closed Hamel basis. Further we investigate the existence of certain c...