# proximity generated by uniformity

Definition. Let $X$ be a uniform space with uniformity $\mathcal{U}$. The uniform proximity, or proximity generated by $\mathcal{U}$, is a binary relation $\mathrel{\delta}$ on the subsets of $X$, given by the formula

 $A\mathrel{\delta}B\qquad\iff\qquad\forall U\in\mathcal{U}:U\cap(A\times B)\neq\emptyset.$

Correctness of the above proximity is given by the below theorem:

Theorem. Proximity generated by a uniformity is a proximity. In other words, $X$ is a proximity space with proximity $\mathrel{\delta}$.

Title proximity generated by uniformity ProximityGeneratedByUniformity 2013-03-22 16:56:20 2013-03-22 16:56:20 porton (9363) porton (9363) 9 porton (9363) Definition msc 54E17 msc 54E15 msc 54E05 uniform proximity