Let $X$ be a topological space, $\mathfrak{S}$ a cover of $X$ \and $C_b(X,\mathbb{K};\mathfrak{S})$ the algebra of all $\mathbb{K}$-valued continuous functions on $X$, which are bounded on every $S \in \mathfrak{S}$. Necessary and sufficient conditions for a subalgebra $\mathcal{A}$ of $C_b(X,\mathbb{K};\mathfrak{S})$ to be dense in $C_b(X,\mathbb{K};\mathfrak{S})$ in the topology $\tau_\mathfrak{S}$ of $\mathfrak{S}$-convergence and in the $\mathfrak{S}$-strict topology $\beta_\mathfrak{S}$ on $C_b(X,\mathbb{K};\mathfrak{S})$ are given. Also, necessary and sufficient conditions for the completeness of the topological algebras $(C_b(X,\mathbb{K};\mathfrak{S}),\tau_{\mathfrak{S}})$ and $(C_b(X,\mathbb{K};\mathfrak{S} ),\beta_{\mathfrak{S}})$ are given.