Remarks on density topology and its category analogue
On Some Geometrical Characterization of Singular Normed Measures
Two remarks about surfaces
Upper and lower limits with respect to the m-ideal
On points of absolute continuity. Functions of several variables
On piecewise flat surfaces in the sense of Toralballa
Qualitative cluster sets
Superpositions of transformations of bounded variation
On the family of sets of approximate limit numbers
Interesting sets which were constructed by Polish mathematicians
This paper describes several examples of sets having interesting and sometimes unexpected properties which arose thanks to the ingenuity of Polish mathematicians. A considerable number of these constructions are due to Wacªaw Sierpi«ski, the remaining ones come from Kazimierz Kuratowski, Bronisªaw Knaster, Stefan Mazurkiewicz, Stanisław Ruziewicz, Otto Nikodym, Zenon Waraszkiewicz and Herman Auerbach.
Functions or functional sequences with special properties whose constructions we owe to Polish mathematicians
In the scholarly output of Polish mathematicians of the 20th century one can find numerous examples of functions and sequences of functions with unusual or unexpected properties. This paper discusses certain results of Wacław Sierpiński, Stanisław Saks, Stefan Mazurkiewicz, Hugo Steinhaus, Stefan Banach, Witold Wilkosz, Stanisław Ruziewicz, Antoni Zygmund, Józef Marcinkiewicz, Zygmunt Zahorski, Zbigniew Grande and Jan Lipiński. Some constructions are presented in details.
The work of Stanisław Ruziewicz
Stanislaw Ruziewicz was born in 1889. He studied at the Jan Kazimierz University in Lvov in the years 1908-1913. Doctorate under the direction of J. Puzyna received in Lvov in 1912, received his Habilitation in 1918. The associate professor was in 1920, and the ordinary professorship in 1924. He got chair position on one of four mathematics departments at the Jan Kazimierz University (other chairs of departments were Eustachy Żyliński, Hugo Steinhaus and Stefan Banach).
Zygmunt Zahorski's works on the first derivative
Zygmunt Zahorski was born in 1914. In 1932 he began his studies at the Faculty of Mechanics, the Warsaw Polytechnic intending to study aeronautical engineering. Two years later he joined to the Faculty of Mathematics at Warsaw University for parallel study (in slow motion) where he included two full years and, in the same time, a part of the third year course at the Polytechnic. The mathematical studies he completed in 1939. He published his first work in 1937. In the same year he started working...
Waclaw Sierpinski's works on real functions
The enormous scientific achievements of Wacław Sierpiński the subject of real functions occupies an important place. Overview of the most important results in this field can be found in the article by Stanislaw Hartman reproduced in Selected Works of Wacław Sierpiński in Volume II, pages 25-31 (Choisies Oeuvres, Volume III, PWN 1975). The purpose of this article is to draw attention also to some works which prof. Hartman does not mention, and to show how, in some cases Professor's results inspired...
Results of Polish mathematicians concerning the Baire classification.
The paper describes results of Polish mathematicians of XX century concerning the Baire classification of functions and Borel classification of sets. As usual, numerous theorems came from Wacław Sierpiński, remaining are due to Kazimierz Kuratowski, Stefan Banach, Stefan Kempisty, Edward Szpilrajn (Marczewski), Andrzej Alexiewicz, Władysław Orlicz, Adolf Lindenbaum, Roman Sikorski, Ryszard Engelking, Włodzimierz Holsztyński, Tadeusz Traczyk, Janina Staniszewska, Zygmunt Zahorski and Samuel Eilenberg....
Remark on the theorem of Egoroff
A category Ψ-density topology
Ψ-density point of a Lebesgue measurable set was introduced by Taylor in [Taylor S.J., On strengthening the Lebesgue Density Theorem, Fund. Math., 1958, 46, 305–315] and [Taylor S.J., An alternative form of Egoroff’s theorem, Fund. Math., 1960, 48, 169–174] as an answer to a problem posed by Ulam. We present a category analogue of the notion and of the Ψ-density topology. We define a category analogue of the Ψ-density point of the set A at a point x as the Ψ-density point at x of the regular open...
Convergence almost everywhere of sequences of measurable functions
Multiplying balls in the space of continuous functions on [0,1]
Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ +, min, max then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ +,·,min,max, then the set is residual whenever E is residual in...
On homeomorphisms of the density type topologies
This paper is dealing of the homeomorphisms of the density type topologies introduced in [3].
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