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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,
Topological speedups of ℤd-actions
Published in Dynamical Systems, 2022
Aimee S. A. Johnson, David M. McClendon
Step 5: Copy the refinement of Step 4 over to. Recall that we had, for each α, Define, for each α, the sets ; these are the atoms of contained in the atom which belongs to the base of . Thus gives different notation for the refinement constructed in Step 4. Then choose disjoint clopen subsets , whose union is all of , with . Denote by the castle refinement of over these partitions; we now have (5c) and (5d) of the induction.
Induced hyperspace dynamical systems of symbolic dynamical systems
Published in International Journal of General Systems, 2018
Zhiming Li, Minghan Wang, Guo Wei
As 's are open (in fact clopen) subsets of , it follows from the definition of the Vietoris topology that is an open subset of the hyperspace . Moreover, every element of is a closed subset of , thus a compact subset as is compact; is a minimal open cover of . Hence, is a minimal open cover of the hyperspace and the elements of are disjoint.
Rank 2 proximal Cantor systems are residually scrambled
Published in Dynamical Systems, 2018
There are positive integers, rn, l(n, 1),… , l(n, rn), and closed and open sets, Z(n, i, j), i = 1, … rn, 0 ≤ j < l(n, i) for each n = 1, 2, 3,… , which satisfy the following: the sequence, , is a decreasing sequence of clopen sets with intersection { x0 },for each n, is a partition of X, which satisfies f(Z(n, i, j)) = Z(n, i, j + 1) for 0 ≤ j < l(n, i) − 1,for all n, is a refinement of , generates the topology of X.