Definition 3

Let $A$ be a subset of a uniform space $X$. The uniformity on $A$ whose set of entourages is the trace on $A \times A$ of the set of entourages of $X$ is called the uniformity induced on $A$ by the uniformity of $X$.

The topology induced by the uniformity induced on $A$ is the same as the topology induced on $A$ by the topology of $X$; the set $A$, together with the uniformity and the topology induced by those of $X$, is called a uniform subspace of $X$.

If $A$ is a subset of a uniform space $X$ and if $f : X \to X'$ is a uniformly continuous mapping, then the restriction $f|A$ is a uniformly continuous mapping of $A$ into $X'$. If $A' \subset X'$ is such that $f(X) \subset A'$, then the mapping of $X$ into the uniform subspace $A'$ of $X'$, having the same graph as $f$, is again uniformly continuous (no. 3, Proposition 4).

If $B \subset A \subset X$, then the uniform subspace $B$ of $X$ is identical with the uniform subspace $B$ of the uniform subspace $A$ of $X$ (transitivity of induced uniform structures; no. 3, Proposition 5).