Let $B_k \subset {[0,1]^{k+1}}$ and define the map: $$ \phi_k:B_k\mapsto C^k[0,1]:(\beta_0,\beta_1, \ldots,\beta_k)\mapsto\sum_{i=0}^k\beta_i b_{i,k}, $$ where $\{b_{i,k}(t)=\binom{k}{i}t^i(1-t)^{k-1},\, t\in[0,1], i=0,\ldots,k\}$ denotes the Bernstein Polynomial basis of degree $k$. Denote $P_k:=\{f\in C^k[0,1]: f=\phi_k(\beta_0, \ldots,\beta_k), (\beta_0, \ldots,\beta_k)\in B_k\}$. Endow $B_k$ and $P_k$ with the Euclidean and the uniform metrics, respectively, denoted by $d_E$ and $d_\infty$. Then, it can be readily seen that the map $\phi_k:(B_k,d_E)\mapsto (P_k, d_\infty)$ is continuous.
Now, let me be not very precise for a moment and define a general map $$\phi:\cup_{k=1}^\infty B_k\mapsto \cup_{k=1}^\infty P_k$$ such that, if $\boldsymbol{\beta}\in B_k$, then $\phi(\boldsymbol{\beta})=\phi_k(\boldsymbol{\beta})$. Which type of (metric) topology shoud I use on the "union spaces" so that the map $\phi$ is continuous? Should I consider $\phi$ as a map between the co-product topological spaces for both $(B_k,d_E)$'s and $(P_k,d_\infty)$? Or it is sufficient to consider the co-product topological space for $(B_k,d_E)$'s and endow $\cup_{k=1}^\infty P_k$ with the uniform metric?