Example 2

On any set $X$, the topology induced by the discrete uniformity (no. 1, Example 2) is the discrete topology.

In the future, when we speak of the topology of a uniform space $X$, we shall always mean the topology induced by the uniform structure of the space, unless the contrary is expressly stated. The topological space obtained by putting this topology on the set $X$ is sometimes called the topological space underlying the uniform space in question. For example, when we say that a uniform space is Hausdorff, or compact, or locally compact, etc., we mean that the underlying topological space has this property.

If $X$ and $X'$ are two uniform spaces, any isomorphism $f$ of the uniform structure of $X$ onto that of $X'$ is also a homeomorphism of $X$ onto $X'$; we say that $X$ is an isomorphism of the uniform space $X$ onto the uniform space $X'$. It should be noted that a homeomorphism of $X$ onto $X'$ is not necessarily an isomorphism of the uniform structure of $X$ onto that of $X'$.

In other words, distinct uniformities on the same set $X$ can induce the same topology. *For example, on ]0, +∞[, the additive and the multiplicative uniformities (which are distinct : Chapter III, § 6, Exercise 17) induce the same topology.* For another example see § 2, no. 2, Remark 1.