The Alexandroff property and the preservation of strong uniform continuity

Gerald Beer


In this paper we extend the theory of strong uniform continuity and strong uniform convergence, developed in the setting of metric spaces in, to the uniform space setting, where again the notion of shields plays a key role. Further, we display appropriate bornological/variational modifications of classical properties of Alexandroff [1] and of Bartle for nets of continuous functions, that combined with pointwise convergence, yield continuity of the limit for functions between metric spaces.


Strong uniform continuity; Strong uniform convergence; Preservation of continuity; Variational convergence; Bornology; The Alexandroff property; The Bartle property; Shield; Quasi-uniform convergence; Bornological uniform cover; Sticking topology

Full Text:



P. Alexandroff, Einf¨uhring in die Mengenlehre und die theorie der rellen Funktionen, Deutscher Verlag der Wissenschaften, Berlin, 1964

C. Arzelà, Intorno alla continuitá della somma di infinitá di funzioni continue, Rend. R. Accad. Sci. Bologna (1883-84), 79-84.

C. Arzelà, Sulle serie di funzioni, Mem. R. Accad. Sci. Ist. Bologna, serie 5 (8) (1899-1900), 131-186 and 701-744.

H. Attouch, R. Lucchetti, and R. Wets, The topology of the -Hausdorff distance, Ann. Mat. Pura Appl. 160 (1991), 303-320.

H. Attouch and R. Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), 695-730.

M.Atsuji, Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958), 11-16.

R. Bartle, On compactness in functional analysis, Trans. Amer. Math. Soc. 79 (1955), 35-57.

G. Beer, Topologies on closed and closed convex sets, Kluwer Acad. Publ., Dordrecht, 1993.

G. Beer, C. Costantini, and S. Levi, Bornological convergence and shields, preprint.

G. Beer, C. Costantini, and S. Levi, Total boundedness in metrizable spaces, Houston J. Math., to appear.

G. Beer and A. Di Concilio, Uniform continuity on bounded sets and the Attouch-Wets topology, Proc. Amer. Math. Soc. 112 (1991), 235-243.

G. Beer and S. Levi, Pseudometrizable bornological convergence is Attouch-Wets convergence , J. Convex Anal. 15 (2008), 439-453.

G. Beer and S. Levi, Strong uniform continuity, J. Math. Anal. Appl. 350 (2009), 568-589.

G. Beer and S. Levi, Uniform continuity, uniform convergence, and shields, Set-Valued and Variational Anal. 18 (2010), 251–275.

G. Beer and M. Segura, Well-posedness, bornologies, and the structure of metric spaces, Appl. Gen. Top. 10 (2009), 131-157.

N. Bouleau, Une structure uniforme sur un espace F(E, F), Cahiers Topologie Géom. Diff., 11 (1969), 207-214.

N. Bouleau, On the coarsest topology preserving continuity, preprint, 2006.

A. Caserta, G. Di Maio and L. Holá, Arzelà’s theorem and strong uniform convergence on bornologies, J. Math. Anal. Appl. 371 (2010), 384-392.

N. Dunford and J. Schwartz, Linear operators part I, Wiley Interscience, New York, 1988

H. Hogbe-Nlend, Bornologies and functional analysis, North-Holland, Amsterdam, 1977.

S.-T. Hu, Boundedness in a topological space, J. Math Pures Appl. 228 (1949), 287-320.

R. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, Springer Verlag, Berlin, 1988.

J. Rainwater, Spaces whose finest uniformity is metric, Pacific J. Math 9 (1959), 567-570.

S. Willard, General topology, Addison-Wesley, Reading, MA, 1970.

Abstract Views

Metrics Loading ...

Metrics powered by PLOS ALM

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147