scholarly journals On the difference property of Borel measurable functions

2010 ◽  
Vol 208 (1) ◽  
pp. 57-73
Author(s):  
Hiroshi Fujita ◽  
Tamás Mátrai
Keyword(s):  
Difference Property ◽  
Measurable Functions ◽  
The Difference ◽  
Borel Measurable

scholarly journals Some Applications of Supra Preopen Sets

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
M. E. El-Shafei ◽  
A. H. Zakari ◽  
T. M. Al-shami
Keyword(s):  
Topological Spaces ◽  
Difference Property ◽  
Boundary Points ◽  
Separation Axioms ◽  
The Difference

The aim of this work is to define some concepts on supra topological spaces using supra preopen sets and investigate main properties. We started this paper by correcting some results obtained in previous study and presenting further properties of supra preopen sets. Then, we introduce a concept of supra prehomeomorphism maps and discuss its main properties. After that we explore the concepts of supra limit and supra boundary points of a set with respect to supra preopen sets and examine their behaviours on the spaces that possess the difference property. Finally, we formulate the concepts of supra pre-Ti-spaces i=0,1,2,3,4 and give completely descriptions for each one of them. In general, we study their main properties in detail and show the implications of these separation axioms among themselves as well as with STi-space with the help of some interesting examples.

scholarly journals Pointwise measurable functions

2004 ◽  
Vol 95 (2) ◽  
pp. 305
Author(s):  
Herman Render ◽  
Lothar Rogge
Keyword(s):  
Large Class ◽  
Metric Space ◽  
Measurable Function ◽  
Polish Space ◽  
Topological Spaces ◽  
Range Space ◽  
Compact Metric Space ◽  
Delta Set ◽  
Measurable Functions ◽  
Borel Measurable

We introduce the new concept of pointwise measurability. It is shown in this paper that a measurable function is measurable at each point and that for a large class of topological spaces the converse also holds. Moreover it can be seen that a function which is continuous at a point is Borel-measurable at this point too. Furthermore the set of measurability points is considered. If the range space is a $\sigma$-compact metric space, then this set is a $G_{\delta}$-set; if the range space is only a Polish space this is in general not true any longer.

scholarly journals On the Dirichlet Problem in the Axiomatic Theory of Harmonic Functions

1963 ◽  
Vol 23 ◽  
pp. 73-96 ◽  
Author(s):  
N. Boboc ◽  
C. Constantinescu ◽  
A. Cornea
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Minimum Principle ◽  
Continuous Functions ◽  
Axiomatic Theory ◽  
Measurable Functions ◽  
Bounded Functions ◽  
Compact Riemann Surfaces ◽  
Borel Measurable

In the frame of the recent axiomatic theories of harmonic functions [2], [3], [1], it has been shown that the continuous bounded functions on the boundaries of relatively compact open sets are resolutive [5], [1]. The aim of the present paper is to substitute in these results the continuous functions by Borel-measurable functions and to leave out the restriction that the open sets are relatively compact. H. Bauer has replaced the axiom 3 of Brelot’s axiomatic by two weaker axioms: the axiom of separation (Trennungsaxiom) and the axiom K1. Since the axiom of separation is not fulfilled in some important cases (e.g. the compact Riemann surfaces) we shall weaken this axiom too, substituting it by one of its consequences: the minimum principle for hyperharmonic functions.

CHIRAL POTTS MODEL: CORNER TRANSFER MATRICES AND PARAMETRIZATIONS

1993 ◽  
Vol 07 (20n21) ◽  
pp. 3489-3500 ◽  
Author(s):  
R.J. BAXTER
Keyword(s):  
Transfer Matrix ◽  
Potts Model ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Weight Functions ◽  
Chiral Potts Model ◽  
Transfer Matrices ◽  
Corner Transfer Matrices ◽  
Difference Property ◽  
The Difference

We consider the star-triangle relation and the form of its solutions. We present some simple parametrizations of the weight functions of the three-state chiral Potts model. This model does not have the “difference property”: we discuss the resulting difficulties in attempting to use the corner transfer matrix method for this model.

scholarly journals Limit Points and Separation Axioms with Respect to Supra Semi-open Sets

2020 ◽  
Vol 13 (3) ◽  
pp. 427-443 ◽  
Author(s):  
Tareq M. AL-shami ◽  
E. A. Abo-Tabl ◽  
Baravan Assad ◽  
Mohamed Arahet
Keyword(s):  
Real Life ◽  
Research Area ◽  
Topological Spaces ◽  
Difference Property ◽  
Separation Axioms ◽  
Life Problems ◽  
New Concepts ◽  
Limit Points ◽  
The Difference

Sometimes we need to minimize the conditions of topology for different reasons such as obtaining more convenient structures to describe some real-life problems, or constructing some counterexamples whom show the interrelations between certain topological concepts, or preserving some properties under fewer conditions of those on topology. To contribute this research area, in this paper, we establish some new concepts on supra topological spaces using supra semi-open sets and give some characterizations of them. First, we introduce a concept of supra semi limit points of a set and study main properties, in particular, on the spaces that possess the difference property. Second, we define and investigate new separation axioms, namely supra semi Ti-spaces (i = 0, 1, 2, 3, 4) and give complete descriptions for each one of them. We provide some examples to show the relationships between them as well as with STi-space.

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