Recurrent Transformation Groups

1969 ◽  
Vol 21 ◽  
pp. 564-575
Author(s):  
R. A. Christiansen
Keyword(s):  
Topological Space ◽  
Transformation Groups ◽  
Sufficient Condition ◽  
Almost Periodic ◽  
Compact Topological Space ◽  
Trivial Subgroup

Let (X, T, π) denote a flow, whereXis a compact topological space metrizable byd, andTis a closed non-trivial subgroup of the reals under addition.Tisrecurrentif and only if for eachands> 0, there existst>ssuch thatx∈Ximplies. IfTis almost-periodic, thenTis both recurrent and distal. In§§4 and 5, it is shown that, under more stringent hypotheses, the recurrence ofTis neither a necessary nor a sufficient condition forTto be distal. LetSbe a closed non-trivial subgroup ofT. It is shown in§3 thatTis recurrent if and only ifSis recurrent. From this result, we obtain a solution to a problem posed by Nemyckiĭ (16, p. 492, Problem 6).

Order continuous measures

1964 ◽  
Vol 60 (2) ◽  
pp. 205-207 ◽  
Author(s):  
G. T. Roberts
Keyword(s):  
Topological Space ◽  
Linear Form ◽  
Vector Lattice ◽  
Measure Zero ◽  
Continuous Functions ◽  
Order Convergence ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Continuous Measures ◽  
Order Continuous

1. Objective. It is possible to define order convergence on the vector lattice of all continuous functions of compact support on a locally compact topological space. Every measure is a linear form on this vector lattice. The object of this paper is to prove that a measure is such that every set of the first category of Baire has measure zero if and only if the measure is a linear form which is continuous in the order convergence.

Separable Topological Space of Hereditary

NUTA Journal ◽  
2020 ◽  
Vol 7 (1-2) ◽  
pp. 68-70
Author(s):  
Raj Narayan Yadav ◽  
Bed Prasad Regmi ◽  
Surendra Raj Pathak
Keyword(s):  
Topological Space ◽  
Topological Property ◽  
Sufficient Condition ◽  
Topological Properties ◽  
Necessary And Sufficient Condition ◽  
Separable Space ◽  
Separable Topological Space ◽  
Necessary And Sufficient

A property of a topological space is termed hereditary ifand only if every subspace of a space with the property also has the property. The purpose of this article is to prove that the topological property of separable space is hereditary. In this paper we determine some topological properties which are hereditary and investigate necessary and sufficient condition functions for sub-spaces to possess properties of sub-spaces which are not in general hereditary.

Construction of sheaves by tolerance relations

2016 ◽  
Vol 09 (02) ◽  
pp. 1650035 ◽  
Author(s):  
M. P. K. Kishore ◽  
R. V. G. Ravi Kumar ◽  
P. Vamsi Sagar
Keyword(s):  
Topological Space ◽  
Universal Algebra ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

In this paper a method to construct a global sheaf space over a topological space for an arbitrary set using tolerance relations is proposed. It is observed that in general, the sheaf constructed by this method is different from the sheaf constructed by the method discussed in [U. M. Swamy, Representation of Universal algebra by sheaves, Proc. Amer. Math. Soc. 45 (1974) 55–58]. A necessary and sufficient condition for embedding a non-empty set into the set of all global sections of a sheaf over given topological space is established, and further an application over graphs is studied.

scholarly journals On locally weakly almost periodic transformation groups

1982 ◽  
Vol 25 (2) ◽  
pp. 215-219
Author(s):  
Saber Elaydi
Keyword(s):  
Phase Space ◽  
Transformation Group ◽  
Transformation Groups ◽  
Almost Periodic ◽  
Locally Compact ◽  
Weakly Almost Periodic ◽  
Phase Group ◽  
Periodic Transformation

It is shown that a transformation group with a locally compact Hausdorff phase space and an abelian phase group is locally weakly almost periodic if and only if it is P-locally weakly almost periodic for some replete semigroup P in the phase group.

scholarly journals Products of a C-Measure and a Locally Integrable Mapping

1957 ◽  
Vol 9 ◽  
pp. 475-486 ◽  
Author(s):  
Marston Morse ◽  
William Transue
Keyword(s):  
Integral Representation ◽  
Topological Space ◽  
Orlicz Spaces ◽  
Complex Numbers ◽  
Bounded Operators ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Special Cases ◽  
Integrable Mapping

Let C be the field of complex numbers and E a locally compact topological space. The authors' theory of C-bimeasures Λ and their Λ-integrals in (1; 2) leads to integral representation of bounded operators from A to B' where A and B are MT-spaces as defined in (3). These MT-spaces include the -spaces and Orlicz spaces as special cases.

On product sets in a unimodular group

1970 ◽  
Vol 68 (2) ◽  
pp. 355-357
Author(s):  
M. McCrudden
Keyword(s):  
Topological Space ◽  
Convergent Subsequence ◽  
Unimodular Group ◽  
Prove Theorem ◽  
Compact Topological Space ◽  
Corrected Proof ◽  
Product Sets

It has been pointed out to me by Professor A. M. Macbeath that the proof of Theorem 2 which is given in (2) is incorrect, in that it relies on Proposition 3·1 of (2), which is false, the mistake in its proof being the assumption that in an arbitrary compact topological space, every sequence has a convergent subsequence. However, we can still prove Theorem 2 of (2), simply by replacing section 3 of (2) by the corrected section 3 below, and replacing the proof of Proposition 4·2 of (2) by the corrected proof of Proposition 4·2 given below.

ON HAUSDORFF DIMENSION AND TOPOLOGICAL ENTROPY

Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 363-370 ◽  
Author(s):  
DONGKUI MA ◽  
MIN WU
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Topological Space ◽  
Topological Entropy ◽  
Continuous Map ◽  
Compact Topological Space ◽  
Metric Function

Let f: X → X be a continuous map of a compact topological space. If there exists a metric function on X and it satisfies some restricted conditions, we obtain some relationships between Hausdorff dimension and topological entropy for any Z ⊆ X. Using those results, we also obtain a variational principle of dimensions, generalize some known results and give some examples.

scholarly journals Functional Space C(ω), C0(ω)

2012 ◽  
Vol 20 (1) ◽  
pp. 15-22
Author(s):  
Katuhiko Kanazashi ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Topological Space ◽  
Banach Algebra ◽  
Function Space ◽  
Normed Space ◽  
Functional Space ◽  
Continuous Functions ◽  
Bounded Support ◽  
Compact Topological Space ◽  
Definition Of ◽  
Complex Valued

Functional Space C(ω), C0(ω) In this article, first we give a definition of a functional space which is constructed from all complex-valued continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all complex-valued continuous functions with bounded support. We also prove that this function space is a complex normed space.

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