Some remarks about two definitions of continuity

• Autorzy: Cichoń J.
• Języki publikacji: EN

Abstrakty

EN

We formulate some variant of the axiom of choice which is neccessary and sufficient for the equivalence ofthe two definitions of continuity (the Cauchy and the Heine definition) for functions from the real line into any metric space. This result definitely solves the problem of the equivalence of these two classical definitions. We also slightly improve one of Sierpiński's results about global continuity of functions from the real line. We give a negative answer to the problem from Jaegermann's ciassical paper [2].

Słowa kluczowe

EN

axiom of choice; continuity; sequential continuity; metric spaces

PL

aksjomat wyboru; ciągłość; ciągłość sekwencyjna; przestrzeń metryczna

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2003
• Tom: Vol. 51, no 1
• Strony: 43--47
• Opis fizyczny: Bibliogr. 4 poz.

Twórcy

autor

Cichoń J.

• Institute of Mathematics, Technical University of Wrocław, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland,

Bibliografia

• [1] N. Brunner, Sequential continuity, Kyungpook Math. J., 22 (2) (1982) 98.
• [2] M. Jeagermann, The axiom of choice and two definitions of continuity, Bull. Ac. Pol. Sci., 13 (1965) 699-704.
• [3] T. Jech, Lectures in Set Theory with particular emphasis on the method of forcing, Springer-Verlag, Berlin Heidelberg New York 1971.
• [4] W. Sierpiński, L’axiome de M. Zermelo et son rôle dans la Théorie des Ensembles et l’Analyse, Bull. Intern. Acad. Sci. Cracovie A, (1918) 97-152.