Some density results for uniformly continuous functions

✍ Scribed by I. Garrido; F. Montalvo

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 222 KB
• Volume: 77
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(96)00137-x

✦ Synopsis

Let X be a set and ~ a family of real-valued functions (not necessarily bounded) on X. We denote by/zsX the space X endowed with the weak uniformity generated by ~, and by ~(#~-X) the collection of uniformly contipuous functions ~' o the real line ~.

In this note we study necessary and sufficient conditions in order that the family ~ be uniformly dense in Lt(t~,~X). Firstly, we give a more direct proof of a result by Hager involving an external condition over ~ given in terms of composition with the uniformly continuous and real-valued functions defined on subsets of ~". From this external condition we can derive as easy corollaries most of the results already known in this context. In the second part of this note we obtain an internal necessary and sut~cient condition of uniform density set by means of cerlain covers of X by cozero-sets of functions in ~.