scholarly journals Complete regularity of Ellis semigroups of -actions

2020 ◽  
pp. 1-17
Author(s):  
MARCY BARGE ◽  
JOHANNES KELLENDONK
Keyword(s):  
Complete Regularity ◽  
Completely Regular ◽  
Ellis Semigroup ◽  
Totally Disconnected

Abstract It is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

scholarly journals Invariant metrics on free topological groups

1973 ◽  
Vol 9 (1) ◽  
pp. 83-88 ◽  
Author(s):  
Sidney A. Morris ◽  
H.B. Thompson
Keyword(s):  
Topological Group ◽  
Invariant Metrics ◽  
Regular Space ◽  
Discrete Topology ◽  
Group Topology ◽  
Topological Groups ◽  
Abstract Group ◽  
Completely Regular ◽  
Free Topological Group ◽  
Totally Disconnected

For a completely regular space X, G(X) denotes the free topological group on X in the sense of Graev. Graev proves the existence of G(X) by showing that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). It is natural to ask if the topology given by these two-sided invariant pseudo-metrics on G(X) is precisely the free topological group topology on G(X). If X has the discrete topology the answer is clearly in the affirmative. It is shown here that if X is not totally disconnected then the answer is always in the negative.

On the Complete Regularity of Some Category Spaces

1975 ◽  
Vol 17 (5) ◽  
pp. 651-656 ◽  
Author(s):  
W. Eames
Keyword(s):  
Topological Space ◽  
Density Function ◽  
Measure Space ◽  
Baire Property ◽  
Complete Regularity ◽  
Completely Regular ◽  
Upper Density ◽  
Covering Class ◽  
Sequential Covering ◽  
Null Set

A category space is a measure space which is also a topological space, the measure and the topology being related by ‘a set is measurable iff it has the Baire property’ and ‘a set is null iff it is nowhere dense’ [4]. We considered some category spaces in [3]; now we show that if a null set is deleted from the space, then the topology can be taken to be completely regular. The essential part of the construction consists of obtaining a suitable refinement of the original sequential covering class and using the consequent strong upper density function to define the required topology. Then the complete regularity follows much as in [1].

Short Proof of an Internal Characterization of Complete Regularity

1984 ◽  
Vol 27 (4) ◽  
pp. 461-462 ◽  
Author(s):  
Harald Brandenburg ◽  
Adam Mysior
Keyword(s):  
Short Proof ◽  
Complete Regularity ◽  
Completely Regular

AbstractA short proof is given of an internal characterization of completely regular spaces due to J. Kerstan.

scholarly journals A remark on free topological groups with no small subgroups

1974 ◽  
Vol 18 (4) ◽  
pp. 482-484 ◽  
Author(s):  
H. B. Thompson
Keyword(s):  
Topological Group ◽  
Free Group ◽  
Regular Space ◽  
Group Topology ◽  
Topological Properties ◽  
Topological Groups ◽  
Abstract Group ◽  
Completely Regular ◽  
Metric Topology ◽  
Totally Disconnected

For a completely regular space X let G(X) be the Graev free topological group on X. While proving G(X) exists for completely regular spaces X, Graev showed that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). The free group topology on G(X) is usually strictly finer than this pseudo-metric topology. In particular this is the case when X is not totally disconnected (see Morris and Thompson [7]). It is of interest to know when G(X) has no small subgroups (see Morris [5]). Morris and Thompson [6] showed that this is the case if and only if X admits a continuous metric. The proof relied on properties of the free group topology and it is natural to ask if G(X) with its pseudo-metric topology has no small subgroups when and only when X admits a continuous metric. We show that this is the case. Topological properties of G(X) associated with the pseudo-metric topology have recently been studied by Joiner [3] and Abels [1].

A Hahn-Banach Theorem in Subbase Convexity Theory

1980 ◽  
Vol 32 (4) ◽  
pp. 804-820 ◽  
Author(s):  
M. van de Vel
Keyword(s):  
Main Tool ◽  
Complete Regularity ◽  
Completely Regular ◽  
Banach Theorem ◽  
Convexity Theory ◽  
Hausdorff Compactifications

In the last fifteen years, topology has shown up with an increasing interest in the use of closed subbases. Starting from Frink's internal characterization of complete regularity (Frink [6]), DeGroot and Aarts used closed subbases to obtain Hausdorff compactifications of completely regular spaces, thus giving a characterization of the latter in terms of their subbases [1]. The main tool of that paper is the notion of a linked system, which naturally leads to the notions of supercompactness and superextensions [7]. After 1970, these two topics developed to indepedennt theories, with several deep results available at this moment. Most results up to 1976 are summarized in [12].

scholarly journals Local invariance of free topological groups

1986 ◽  
Vol 29 (1) ◽  
pp. 1-5 ◽  
Author(s):  
M. S. Khan ◽  
Sidney A. Morris ◽  
Peter Nickolas
Keyword(s):  
Topological Group ◽  
Hausdorff Space ◽  
Convergent Sequence ◽  
Discrete Topology ◽  
Group Topology ◽  
Topological Groups ◽  
Completely Regular ◽  
Free Topological Group ◽  
Hausdorff Group ◽  
Totally Disconnected

In 1948, M. I. Graev [2] proved that the free topological group on a completely regular Hausdorff space is Hausdorff, by showing that the free group admits a certain locally invariant Hausdorff group topology. It is natural to ask if Graev's locally invariant topology is the free topological group topology. If X has the discrete topology, the answer is clearly in the affirmative. In 1973, Morris-Thompson [6] showed that if X is not totally disconnected then the answer is negative. Nickolas [7] showed that this is also the case if X has any (non-trivial) convergent sequence (for example, if X is any non-discrete metric space). Recently, Fay and Smith Thomas handled the case when X has a completely regular Hausdorff quotient space which has an infinite compact subspace (or more particularly a non-trivial convergent sequence).(Fay-Smith Thomas observe that their class of spaces includes some but not all those dealt with by Morris-Thompson.)

The topologies of separate continuity. II

1972 ◽  
Vol 71 (2) ◽  
pp. 307-319 ◽  
Author(s):  
C. J. Knight ◽  
W. Moran ◽  
J. S. Pym
Keyword(s):  
Tensor Product ◽  
Topological Spaces ◽  
Complete Regularity ◽  
Completely Regular ◽  
Categorical Properties

In (6) we studied a topology T on the product set X × Y of two topological spaces X and Y which was defined by the requirement that each mapping from X × Y which was continuous in each variable separately was also continuous in T; we called (X × Y, T) the tensor product of X and Y, and denoted it by X ⊗ Y. Theorem (3·2) of (6) indicated that X ⊗ Y was rarely completely regular; as complete regularity is of importance in analytic problems, we consider here a ‘completely regular tensor product’ . Roughly speaking, gives a tensor product in the category of completely regular topological spaces. The categorical properties of are discussed in section 5.

scholarly journals Some New Variants of Relative Regularity via Regularly Closed Sets

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Sehar Shakeel Raina ◽  
A. K. Das
Keyword(s):  
Topological Property ◽  
Topological Properties ◽  
Complete Regularity ◽  
Completely Regular ◽  
Closed Sets ◽  
Relative Property ◽  
Relative Version ◽  
The Absolute

Every topological property can be associated with its relative version in such a way that when smaller space coincides with larger space, then this relative property coincides with the absolute one. This notion of relative topological properties was introduced by Arhangel’skii and Ganedi in 1989. Singal and Arya introduced the concepts of almost regular spaces in 1969 and almost completely regular spaces in 1970. In this paper, we have studied various relative versions of almost regularity, complete regularity, and almost complete regularity. We investigated some of their properties and established relationships of these spaces with each other and with the existing relative properties.

scholarly journals On the lattice of varieties of completely regular semigroups

1983 ◽  
Vol 35 (2) ◽  
pp. 227-235 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Lattice Of Varieties ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Variety Of Groups ◽  
Completely Regular Semigroups ◽  
Subdirect Products

AbstractSeveral morphisms of this lattice V(CR) are found, leading to decompostions of it, and various sublattices, into subdirect products of interval sublattices. For example the map V → V ∪ G (where G is the variety of groups) is shown to be a retraction of V(CR); from modularity of the lattice V(BG) of varieties of bands of groups it follows that the map V → (V ∪ V V G) is an isomorphism of V(BG).

scholarly journals Dendrogramic Representation of Data: CHSH Violation vs. Nonergodicity

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 971
Author(s):  
Oded Shor ◽  
Felix Benninger ◽  
Andrei Khrennikov
Keyword(s):  
Experimental Data ◽  
Clustering Algorithms ◽  
Realistic Model ◽  
Minkowski Geometry ◽  
Ultrametric Spaces ◽  
Chsh Inequality ◽  
Classical Quantum ◽  
Basic Features ◽  
Totally Disconnected ◽  
Explicate Order

This paper is devoted to the foundational problems of dendrogramic holographic theory (DH theory). We used the ontic–epistemic (implicate–explicate order) methodology. The epistemic counterpart is based on the representation of data by dendrograms constructed with hierarchic clustering algorithms. The ontic universe is described as a p-adic tree; it is zero-dimensional, totally disconnected, disordered, and bounded (in p-adic ultrametric spaces). Classical–quantum interrelations lose their sharpness; generally, simple dendrograms are “more quantum” than complex ones. We used the CHSH inequality as a measure of quantum-likeness. We demonstrate that it can be violated by classical experimental data represented by dendrograms. The seed of this violation is neither nonlocality nor a rejection of realism, but the nonergodicity of dendrogramic time series. Generally, the violation of ergodicity is one of the basic features of DH theory. The dendrogramic representation leads to the local realistic model that violates the CHSH inequality. We also considered DH theory for Minkowski geometry and monitored the dependence of CHSH violation and nonergodicity on geometry, as well as a Lorentz transformation of data.

Sign in / Sign up

Export Citation Format

Share Document