Entourage

Remarks

On $\R$ with the uniformity of the metric $\abs{x - y}$, the set $\set{(x, y) : \abs{x - y} < 1}$ is an entourage, and $U[x] = (x - 1, x + 1)$. The set $\set{(x, y) : \abs{x - y} < 1 / (1 + x^2)}$ contains the diagonal, but it is not an entourage of that uniformity. It contains no set $\set{(x, y) : \abs{x - y} < \varepsilon}$ with $\varepsilon > 0$, because its width tends to $0$ as $\abs{x}$ grows. Every entourage contains a symmetric entourage, namely $U \cap U^{-1}$. The sets $U[x]$, for $U$ an entourage, form a neighbourhood base at $x$ of the uniform topology.