The restriction function is a uniformly continuous mapping

Question

If A is a subset of a uniform space $ X $ and if $ f:X\to X^{'} $ is a uniformly continuous mapping , then the restriction $ f_{A}:A\to X^{'} $ is a uniformly continuous mapping of $ A $ into $ X^{'} $?

Answer 1

This folllows from $$(f_A \times f_A)^{-1}[U] = (f \times f)^{-1}[U] \cap (A \times A)$$ where $U$ is any entourage from $X'$, and the right hand side is in the subspace uniformity on $A$ by definition, as $f$ is uniformly continuous.