Topology induced by Uniformity?

Question

Let $f:(X,U) \to (Y,U^{'})$ is a mapping , then $ f $ is uniformly continuous iff $ U^{''}\subset U $ where $ U^{''} $ is the induced uniform structure on $ X $ by $ f $ .

Answer 1

Your statement is a consequence of the minimality part of the definition of "induced uniform structure", or otherwise a direct consequence of the definition that all sets $\{(f \times f)^{-1}[O]: O \in \mathcal{U}'\}$ is a base for it and this base consists of entourages already in $\mathcal{U}$ if $f$ is uniformly continuous!