The Alexandroff property and the preservation of strong uniform continuity

Abstract

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.

Keywords

• 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