Number of open sets for a topology with a countable basis

1993 ◽  
Vol 83 (3) ◽  
pp. 369-374 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Countable Basis

A Mapping Problem and Jp-Index. I

1970 ◽  
Vol 22 (4) ◽  
pp. 705-712 ◽  
Author(s):  
Masami Wakae ◽  
Oma Hamara
Keyword(s):  
Cyclic Group ◽  
Prime Order ◽  
Transformation Group ◽  
Classical Group ◽  
Cohomology Ring ◽  
Transformation Groups ◽  
Normal Space ◽  
Mapping Problem ◽  
Countable Basis ◽  
Kakutani Theorem

Indices of normal spaces with countable basis for equivariant mappings have been investigated by Bourgin [4; 6] and by Wu [11; 12] in the case where the transformation groups are of prime order p. One of us has extended the concept to the case where the transformation group is a cyclic group of order pt and discussed its applications to the Kakutani Theorem (see [10]). In this paper we will define the Jp-index of a normal space with countable basis in the case where the transformation group is a cyclic group of order n, where n is divisible by p. We will decide, by means of the spectral sequence technique of Borel [1; 2], the Jp-index of SO(n) where n is an odd integer divisible by p. The method used in this paper can be applied to find the Jp-index of a classical group G whose cohomology ring over Jp has a system of universally transgressive generators of odd degrees.

Maximality in effective topology

1983 ◽  
Vol 48 (1) ◽  
pp. 100-112 ◽  
Author(s):  
Iraj Kalantari ◽  
Anne Leggett
Keyword(s):  
Topological Space ◽  
Boolean Algebras ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Mathematical Structures ◽  
Topological Vector Spaces ◽  
Linear Orderings ◽  
Recursively Enumerable ◽  
Countable Basis ◽  
Effective Analysis

In this paper we continue the study of the structure of the lattice of recursively enumerable (r.e.) open subsets of a topological space. Work in this approach to effective topology began in Kalantari and Retzlaff [5] and continued in Kalantari [2], Kalantari and Leggett [3] and Kalantari and Remmel [4]. Studies in effectiveness of results in structures other than integers began with the work of Specker [17] and Lacombe [8] on effective analysis.The renewed activity in the study of the effective content of mathematical structures owes much to Nerode's program and Metakides' and Nerode's [11], [12] work on vector spaces and fields. These studies have been extended by Kalantari, Remmel, Retzlaff, Shore and Smith. Similar studies on the effective content of other mathematical structures have been conducted. These include work on topological vector spaces, boolean algebras, linear orderings etc.Kalantari and Retzlaff [5] began a study of effective topological spaces by considering a topological space with a countable basis ⊿ for the topology. The space X is to be fully effective; that is, the basis elements are coded into ω and the operations of intersection of basis elements and the relation of inclusion among them are both computable. An r.e. open subset of X is then represented as the union of basic open sets whose codes lie in an r.e. subset of ω.

scholarly journals Front Cover: Optimal Pump Shaping for Entanglement Control in Any Countable Basis (Adv. Quantum Technol. 10/2021)

2021 ◽  
Vol 4 (10) ◽  
pp. 2170101
Author(s):  
Nicholas Bornman ◽  
Wagner Tavares Buono ◽  
Michael Lovemore ◽  
Andrew Forbes
Keyword(s):  
Front Cover ◽  
Countable Basis

On bases of closed classes of vector functions of many-valued logic

2017 ◽  
Vol 27 (2) ◽  
Author(s):  
Vladimir A. Taimanov
Keyword(s):  
Functional System ◽  
Countable Basis

AbstractWe consider the functional system of vector functions of many-valued logic with the naturally defined operation of superposition and construct examples of closed classes of special type without a basis and with a countable basis.

Optimal Pump Shaping for Entanglement Control in Any Countable Basis

2021 ◽  
pp. 2100066
Author(s):  
Nicholas Bornman ◽  
Wagner Tavares Buono ◽  
Michael Lovemore ◽  
Andrew Forbes
Keyword(s):  
Countable Basis

scholarly journals Equicontinuity on Harmonic Spaces

1967 ◽  
Vol 29 ◽  
pp. 1-6 ◽  
Author(s):  
Corneliu Constantinescu
Keyword(s):  
Harmonic Functions ◽  
Harmonic Space ◽  
Bounded Set ◽  
Countable Basis ◽  
Weakened Form

G. Mokobodzki proved [5] that on any harmonic space with countable basis satisfying the axioms 1, 2, T+, K0 [2] [1] any equally bounded set of harmonic functions is equicontinuous. P. Loeb and B. Walsh showed [4] that the same property holds on a harmonic space without countable basis, if Brelot’s axiom 3 is fulfilled. The aim of this paper is to generalize these results to a harmonic space X satisfying only the axioms 1, 20, K1, [2] [1] where 20 is a weakened form of axiom 2. As a corollary we get: if any point of X possesses two open neighbourhoods U, V such that the set of harmonic functions on U separates the points of U∩V, then X has locally a countable basis.

scholarly journals Maps on D1 and D2 spaces

1984 ◽  
Vol 37 (1) ◽  
pp. 106-109
Author(s):  
H. B. Potoczny
Keyword(s):  
Separation Theorem ◽  
Countable Base ◽  
Closed Set ◽  
Countable Basis

AbstractA space X is said to be D1 provided each closed set has a countable basis for the open sets containing it. It is said to be D2 provided there is a countable base {Un} such that each closed set has a countable base for the open sets containing it, which is a subfamily of {Un}. In this paper, we give a separation theorem for D1 spaces, and provide a characterization of D1 and D2 spaces in terms of maps.

scholarly journals Completely superharmonic measures for the infinitesimal generator A of a diffusion semi-group and positive eigen elements of A

1981 ◽  
Vol 83 ◽  
pp. 53-106 ◽  
Author(s):  
Masayuki Itô ◽  
Noriaki Suzuki
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Infinitesimal Generator ◽  
Hausdorff Space ◽  
Inductive Limit ◽  
Semi Group ◽  
Radon Measures ◽  
Countable Basis ◽  
Inductive Limit Topology ◽  
Locally Compact Hausdorff Space

Let X be a locally compact Hausdorff space with countable basis. We denote byM(X) the topological vector space of all real Radon measures in X with the vague topology,MK(X) the topological vector space of all real Radon measures in X whose supports are compact with the usual inductive limit topology.

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