Corollary 2

Let X be a topological space and Y a metric space. Then the space of all bounded continuous mappings of X into Y is both open and closed in $C_u(X; Y)$; it is complete if Y is complete.

The space in question is $\mathscr{B}(X; Y) \cap C_u(X; Y)$; the first assertion follows from Proposition 2; the second follows from the first ($§ 1$, no. 6, Theorem 2, Corollary 1).