Topic
Clopen set
About: Clopen set is a research topic. Over the lifetime, 389 publications have been published within this topic receiving 3985 citations.
Papers published on a yearly basis
Papers
TL;DR: The main result is that all Borei sets are Ramsey, which was discovered independently by Andrzej Ehrenfeucht, Paul Cohen, and probably many others, but no proof has been published.
Abstract: Definition 1. For a set S and a cardinal κ,In particular, 2ω denotes the power set of the natural numbers and not the cardinal 2ℵ0. We regard 2ω as a topological space with the usual product topology.Definition 2. A set S ⊆ 2ω is Ramsey if there is an M ∈ [ω]ω such that either [M]ω ⊆ S or else [M]ω ⊆ 2ω − S.Erdos and Rado [3, Example 1, p. 434] showed that not every S ⊆ 2ω is Ramsey. In view of the nonconstructive character of the counterexample, one might expect (as Dana Scott has suggested) that all sufficiently definable sets are Ramsey. In fact, our main result (Theorem 2) is that all Borei sets are Ramsey. Soare [10] has applied this result to some problems in recursion theory.The first positive result on Scott's problem was Ramsey's theorem [8, Theorem A]. The next advance was Nash-Williams' generalization of Ramsey's theorem (Corollary 2), which can be interpreted as saying: If S1 and S2 are disjoint open subsets of 2ω, there is an M ∈ [ω]ω such that either [M]ω ⋂ S1 = ∅ or [M]ω ∩ S2 = ⊆. (This is halfway between “clopen sets are Ramsey” and “open sets are Ramsey.”) Then Galvin [4] stated a generalization of Nash-Williams' theorem (Corollary 1) which says, in effect, that open sets are Ramsey; this was discovered independently by Andrzej Ehrenfeucht, Paul Cohen, and probably many others, but no proof has been published.
307 citations
TL;DR: In this article, it was shown that given a linearly recurrent subshift, the set of non-periodic subshift factors is finite up to isomorphism, and a constructive characterization of these subshifts was given.
Abstract: A minimal subshift $(X,T)$ is linearly recurrent if there exists a constant $K$ so that for each clopen set $U$ generated by a finite word $u$ the return time to $U$, with respect to $T$, is bounded by $K|u|$. We prove that given a linearly recurrent subshift $(X,T)$ the set of its non-periodic subshift factors is finite up to isomorphism. We also give a constructive characterization of these subshifts.
236 citations
TL;DR: For almost finite groupoids, this article studied how their homology groups reflect dynamical properties of their topological full groups and showed that the index map induces a homomorphism from H_1 to K_1 of the groupoid C^*-algebra.
Abstract: For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H_0 if and only if there exists an element in the topological full group which maps one to the other. It is also shown that a natural homomorphism, called the index map, from the topological full group to H_1 is surjective and any element of the kernel can be written as a product of four elements of finite order. In particular, the index map induces a homomorphism from H_1 to K_1 of the groupoid C^*-algebra. Explicit computations of homology groups of AF groupoids and etale groupoids arising from subshifts of finite type are also given.
183 citations
TL;DR: Some fundamental properties of the multigranulation rough set model are considered, and it is shown that both the collection of lower definable sets and that of upper definable Set can form a lattice, but such lattices are not distributive, not complemented and pseudo-complemented in the general case.
Abstract: The original rough set model, i.e., Pawlak's single-granulation rough set model has been extended to a multigranulation rough set model, where two kinds of multigranulation approximations, i.e., the optimistic and pessimistic approximations were introduced. In this paper, we consider some fundamental properties of the multigranulation rough set model, and show that (i)Both the collection of lower definable sets and that of upper definable sets in the optimistic multigranulation rough set model can form a lattice, such lattices are not distributive, not complemented and pseudo-complemented in the general case. The collection of definable sets in the optimistic multigranulation rough set model does not even form a lattice in general conditions. (ii)The collection of (lower, upper) definable sets in the optimistic multigranulation rough set model forms a topology on the universe if and only the optimistic multigranulation rough set model is equivalent to Pawlak's single-granulation rough set model. (iii)In the context of the pessimistic multigranulation rough set model, the collections of three different kinds of definable sets coincide with each other, and they determine a clopen topology on the universe, furthermore, they form a Boolean algebra under the usual set-theoretic operations.
135 citations
TL;DR: For almost finite groupoids, this article studied how their homology groups reflect dynamical properties of their topological full groups and showed that the index map induces a homomorphism from H1 to K1 of the groupoid C∗-algebra.
Abstract: For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H0 if and only if there exists an element in the topological full group which maps one to the other. It is also shown that a natural homomorphism, called the index map, from the topological full group to H1 is surjective and any element of the kernel can be written as a product of four elements of finite order. In particular, the index map induces a homomorphism from H1 to K1 of the groupoid C∗-algebra. Explicit computations of homology groups of AF groupoids and etale groupoids arising from subshifts of finite type are also given.
129 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 20 |
| 2020 | 15 |
| 2019 | 12 |
| 2018 | 20 |
| 2017 | 20 |
| 2016 | 14 |