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# 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 $\mathrel{\delta}$.

Defines:

uniform proximity

Keywords:

proximity space, nearness space, uniformity, uniform space

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

54E17*no label found*54E15

*no label found*54E05

*no label found*

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