Uniform topology

The topology generated by a uniform structure. In more detail, let $X$ be a set equipped with a uniform structure (that is, a uniform space) $U$, and for each $x\in X$ let $B(x)$ denote the set of subsets $V(x)$ of $X$ as $V$ runs through the entourages of $U$. Then there is in $X$ one, and moreover only one, topology (called the uniform topology) for which $B(x)$ is the neighbourhood filter at $x$ for any $x\in X$. A topology is called uniformizable if there is a uniform structure that generates it. Not every topological space is uniformizable; for example, non-regular spaces.

Comments

For references see Uniform space.