On the density of the space of continuous and uniformly continuous functions

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. ry 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.