scholarly journals A new approach to ψ-continuity

2009 ◽  
Vol 42 (1) ◽  
pp. 107-117
Author(s):  
Małgorzata Terepeta
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
New Approach ◽  
Density Topology ◽  
The Family ◽  
Real Anal ◽  
Fixed Function

Abstract Let Tψ be a ψ-density topology for a fixed function ψ. For any topological space X with the topology τ we will consider the family C (X, ℝψ) of all continuous functions f from (X, τ) into (ℝ, Tψ). The main aim of this paper is to investigate when C (X, ℝψ) is a ring. This article is based on the results achieved by M. Knox [A characterization of rings of density continuous functions, Real Anal. Exchange 31 (2005), 165-177].

scholarly journals Density topologies on the plane between ordinary and strong. II

2015 ◽  
Vol 62 (1) ◽  
pp. 13-25
Author(s):  
Elżbieta Wagner-Bojakowska ◽  
Władysław Wilczyński
Keyword(s):  
Topological Space ◽  
Measurable Subset ◽  
Continuous Functions ◽  
Density Point ◽  
Density Topology ◽  
Completely Regular ◽  
The Family

Abstract Let C0 denote a set of all non-decreasing continuous functions f : (0, 1] → (0, 1] such that limx→0+f(x) = 0 and f(x) ≤ x for every x ∊ (0, 1], and let A be a measurable subset of the plane. The notions of a density point of A with respect to f and the mapping defined on the family of all measurable subsets of the plane were introduced in Wagner-Bojakowska, E. Wilcziński, W.: Density topologies on the plane between ordinary and strong, Tatra Mt. Math. Publ. 44 (2009), 139 151. This mapping is a lower density, so it allowed us to introduce the topology Tf , analogously to the density topology. In this note, properties of the topology Tf and functions approximately continuous with respect to f are considered. We prove that (ℝ2, Tf) is a completely regular topological space and we study conditions under which topologies generated by two functions f and g are equal.

scholarly journals Ψ-continuous functions and functions preserving ψ-density points

2009 ◽  
Vol 42 (1) ◽  
pp. 175-186
Author(s):  
Małgorzata Filipczak ◽  
Małgorzata Terepeta
Keyword(s):  
Continuous Functions ◽  
Density Topology ◽  
The Family ◽  
Fixed Function

Abstract Let Tψ be the ψ-density topology for a fixed function ψ. We will examine some new properties of the family of ψ-continuous functions (that means continuous functions ƒ: ℝ →ℝ with ψ-density topology Tψ in its domain and range). In the second part of the article we will discuss functions preserving ψ-density points

scholarly journals rc-continuous functions and functions with rc-strongly closed graph

2003 ◽  
Vol 2003 (72) ◽  
pp. 4547-4555
Author(s):  
Bassam Al-Nashef
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Continuous Functions ◽  
Closed Graph ◽  
Compact Spaces ◽  
Extremally Disconnected ◽  
The Family ◽  
Regular Closed

The family of regular closed subsets of a topological space is used to introduce two concepts concerning a functionffrom a spaceXto a spaceY. The first of them is the notion offbeing rc-continuous. One of the established results states that a spaceYis extremally disconnected if and only if each continuous function from a spaceXtoYis rc-continuous. The second concept studied is the notion of a functionfhaving an rc-strongly closed graph. Also one of the established results characterizes rc-compact spaces (≡S-closed spaces) in terms of functions that possess rc-strongly closed graph.

scholarly journals Characterizations of Ideals in IntermediateC-RingsA(X)via theA-Compactifications ofX

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Joshua Sack ◽  
Saleem Watson
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Completely Regular ◽  
Intermediate Ring ◽  
Maximal Ideals ◽  
Kolmogorov Theorem

LetXbe a completely regular topological space. An intermediate ring is a ringA(X)of continuous functions satisfyingC*(X)⊆A(X)⊆C(X). In Redlin and Watson (1987) and in Panman et al. (2012), correspondences𝒵AandℨAare defined between ideals inA(X)andz-filters onX, and it is shown that these extend the well-known correspondences studied separately forC∗(X)andC(X), respectively, to any intermediate ring. Moreover, the inverse map𝒵A←sets up a one-one correspondence between the maximal ideals ofA(X)and thez-ultrafilters onX. In this paper, we define a function𝔎Athat, in the case thatA(X)is aC-ring, describesℨAin terms of extensions of functions to realcompactifications ofX. For such rings, we show thatℨA←mapsz-filters to ideals. We also give a characterization of the maximal ideals inA(X)that generalize the Gelfand-Kolmogorov theorem fromC(X)toA(X).

A theorem about countable decomposability

1982 ◽  
Vol 91 (3) ◽  
pp. 457-458 ◽  
Author(s):  
Roy O. Davies ◽  
Claude Tricot
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Continuous Functions ◽  
Perfect Set ◽  
Baire Function ◽  
Separable Metric Space

A function f:X → ℝ is countably decomposable (into continuous functions) if the topological space X can be partitioned into countably many sets An with each restriction f│ An continuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric space X (see Corollary 1). We deduce that when X is complete they include all functions possessing the property P defined by D. E. Peek in (3): each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has property P.

On the generalization of density topologies on the real line

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Jacek Hejduk ◽  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Density Theorem ◽  
Density Topology ◽  
The Real ◽  
Lebesgue Density ◽  
The Family ◽  
Real Anal ◽  
Similarities And Differences ◽  
Relevant Operator

AbstractThe paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ϕJ generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.

scholarly journals A class of ideals in intermediate rings of continuous functions

2019 ◽  
Vol 20 (1) ◽  
pp. 109 ◽  
Author(s):  
Sagarmoy Bag ◽  
Sudip Kumar Acharyya ◽  
Dhananjoy Mandal
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Alternative Proof ◽  
Prime Ideals ◽  
Borel Sets ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Intermediate Ring ◽  
The Family ◽  
Established Fact

<p>For  any  completely  regular  Hausdorff  topological  space X,  an  intermediate  ring A(X) of  continuous  functions  stands  for  any  ring  lying between C<sup>∗</sup>(X) and C(X).  It is a rather recently established fact that if A(X) ≠ C(X), then there exist non maximal prime ideals in A(X).We offer an alternative proof of it on using the notion of z◦-ideals.  It is realized that a P-space X is discrete if and only if C(X) is identical to the ring of real valued measurable functions defined on the σ-algebra β(X) of all Borel sets in X.  Interrelation between z-ideals, z◦-ideal and Ʒ<sub>A</sub>-ideals in A(X) are examined.  It is proved that within the family of almost P-spaces X, each Ʒ<sub>A</sub> -ideal in A(X) is a z◦-ideal if and only if each z-ideal in A(X) is a z◦-ideal if and only if A(X) = C(X).</p>

scholarly journals A category analogue of the generalization of Lebesgue density topology

2009 ◽  
Vol 42 (1) ◽  
pp. 11-25
Author(s):  
Wojciech Wojdowski
Keyword(s):  
Real Line ◽  
Continuous Functions ◽  
Density Point ◽  
Density Topology ◽  
The Real ◽  
Lebesgue Density ◽  
Real Anal

Abstract . A notion of AI -topology, a generalization of Wilczy´nski’s I-density topology (see [Wilczy´nski, W.: A generalization of the density topology, Real. Anal. Exchange 8 (1982-1983), 16-20] is introduced. The notion is based on his reformulation of the definition od Lebesgue density point. We consider a category version of the topology, which is a category analogue of the notion of an Ad- -density topology on the real line given in [Wojdowski, W.: A generalization ofdensity topology, Real. Anal. Exchange 32 (2006/2007), 1-10]. We also discuss the properties of continuous functions with respect to the topology.

Sign in / Sign up

Export Citation Format

Share Document