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Lattice Theory
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
The Hausdorff condition of separation of points is one of the most frequently discussed and employed condition. All metric spaces are Hausdorff spaces and so are all manifolds. Most importantly, the separation condition implies the uniqueness of limits of sequences and provides a direct connection to Boolean algebra. For example, if X is a Hausdorff space, then according to Definition 2.15(2) ∅ and X are both open and closed sets; also known as clopen sets. Depending on the topology τ, X can have any number of clopen sets. For instance, if X=ℚ with the standard Euclidean topology, then A={x∈ℚ:x2>2} is a clopen set. Note that X is a totally disconnected space with an infinite number of clopen sets. If X is a disconnected Hausdorff space, then its topology contains at least three clopen sets. Now suppose that X is a totally disconnected compact Hausdorff space. Setting A={A⊂X:Aisaclopenset} and defining A+B=(A∩B′)∪(A′∩B)andA·B=A∩B,
Structure and dimension of invariant subsets of expanding Markov maps and joint invariance
Published in Dynamical Systems, 2023
In our setting, we denote the unit interval where 0 and 1 are identified. Let be an expanding Markov map of the circle, i.e. there exist finitely many points such that , and a so that for all , on which f is one-sided differentiable. For such a map we consider to be the set of all compact and non-empty subsets of that are also invariant under the action of f. We endow the set with the Hausdorff metric . M. Urbanski in [17] and C.C. Conley in [3] present some the topological properties of this metric space for expanding Markov maps and for flows respectively. Motivated by that, we further study the topological structure of the metric space . More precisely, we will show that is a compact and totally disconnected metric space.
Topological stability and pseudo-orbit tracing property of Borel measures from the viewpoint of open covers
Published in Dynamical Systems, 2023
In [10], the author introduced the notions of shadowing property and expansivity with the symbolic for a homeomorphism of a non-metrizable compact Hausdorff totally disconnected space by the finite open partition of the involved space. In Definitions 2.5 and 2.9, X is just a compact space but not necessarily Hausdorff or totally disconnected. In addition, it is not hard to prove that the definitions of shadowing property and expansivity of [10] imply Definitions 2.5 and 2.9 in the paper, respectively. In general, the converse is not true (for example, see Example 2.7).
An étale equivalence relation on a tiling space arising from a two-sided subshift and associated C*-algebras
Published in Dynamical Systems, 2021
There is a possiblity that the compact totally disconnected metric space has isolated points. We will discuss a condition for the λ-graph bisystem that the space is a Cantor set in Section 4.4.