On the density of the space of continuous and uniformly continuous functions |
| |
Authors: | Camillo Costantini |
| |
Institution: | Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy |
| |
Abstract: | For X a metrizable space and (Y,ρ) a metric space, with Y pathwise connected, we compute the density of (C(X,(Y,ρ)),σ)—the space of all continuous functions from X to (Y,ρ), endowed with the supremum metric σ. Also, for (X,d) a metric space and (Y,‖⋅‖) a normed space, we compute the density of (UC((X,d),(Y,ρ)),σ) (the space of all uniformly continuous functions from (X,d) to (Y,ρ), where ρ is the metric induced on Y by ‖⋅‖). We also prove that the latter result extends only partially to the case where (Y,ρ) is an arbitrary pathwise connected metric space.To carry such an investigation out, the notions of generalized compact and generalized totally bounded metric space, introduced by the author and A. Barbati in a former paper, turn out to play a crucial rôle. Moreover, we show that the first-mentioned concept provides a precise characterization of those metrizable spaces which attain their extent. |
| |
Keywords: | primary 54C35 54E35 54A25 secondary 54C05 54E40 54E45 |
本文献已被 ScienceDirect 等数据库收录! |
|