On the density of the space of continuous and uniformly continuous functions (Q818349)

In this paper the density of function spaces is discussed. For \(X\) a metrizable space and \((Y, \rho)\) a metric space, with \(Y\) pathwise connected, the author computes the density of the spaces \((C(X, (Y, \rho)), \sigma)\) of all continuous functions, and of the spaces \((UC((X, d), (Y, \rho), \sigma)\) of all uniformly continuous functions endowed with the supremum metric \(\sigma\). To carry out this investigation out, the notions of generalized compact and generalized totally bounded metric space turn out to play an important role.