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Lattice Theory
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
Recall that if X is a topological space and x∈X, then any open set U∈τ with x∈U is a neighborhood of x. If for every pair x,y∈X with x≠y there exist two open sets U and V with x∈U, y∈V and U∩V=∅, then (X,τ) is called a Hausdorff space. For example, the topological space (L,τ), where L is a chain and τ is generated by the intervals basis, is a Hausdorff space.
Introduction to Learning in Games
Published in Hamidou Tembine, Distributed Strategic Learning for Wireless Engineers, 2018
A function between topological spaces is called continuous if the inverse image of every open set is open. A Hausdorff space or separated space is a topological space in which distinct points can have disjoint neighborhoods.
Limsup is needed in the definitions of topological entropy via spanning or separation numbers
Published in Dynamical Systems, 2020
Consider the bijection given by and the function that takes the value when y = z and the value when , where i is such that For any given metric d on A and positive integer κ we can now define a function ρ on as follows: Note that when then will be a metric on that induces the topology of a compact Hausdorff space. To see why the Triangle Inequality holds, note that when , then (18) reduces to the Triangle Inequality for d. If not all are equal, let . When , then ; and when , then . If neither of the former holds, then and .
Topological stability and pseudo-orbit tracing property of Borel measures from the viewpoint of open covers
Published in Dynamical Systems, 2023
In Definition 10 of [5], the authors also introduced the notion of the POTP for a homeomorphism of a locally compact, paracompact and Hausdorff space with the first axiom of countability by the uniform structure which generates the topology of the involved space. In Definition 2.5, X is just a compact space but not necessarily regular or Hausdorff, and so there may not exist a uniform structure which generates the topology even if X is compact. Therefore, we have no idea about the implications of Definition 2.5 and Definition 10 of [5].
A groupoid approach to C*-algebras associated with λ-graph systems and continuous orbit equivalence of subshifts
Published in Dynamical Systems, 2020
Let be a left-resolving λ-graph system over Σ satisfying condition (I). Denote by the associated topological dynamical system. Define the shift space of the topological dynamical system by setting and We endow with the relative topology from the product topology of , so that is a compact Hausdorff space. Hence we have a topological dynamical system with a homeomorphism on the compact Hausdorff space . It is a two-sided extension of . Let be the right one-sided subshift presented by . We then have the shift space and the homeomorphism of the two-sided subshift in a similar way to (23) and (24). The two-sided subshift is written as or Λ for short. For , and with k<l, we set