fundamental system of entourages
Let $(X,\mathcal{U})$ be a uniform space. A subset $\mathcal{B}\subseteq \mathcal{U}$ is a fundamental system of entourages for $\mathcal{U}$ provided that each entourage in $\mathcal{U}$ contains an element of $\mathcal{B}$.
To see that each uniform space $(X,\mathcal{U})$ has a fundamental system of entourages, define
$$\mathcal{B}=\{U\cap {U}^{1}:U\in \mathcal{U}\},$$ 
where ${U}^{1}$ denotes the inverse relation of $U$. Since $\mathcal{U}$ is closed under^{} taking relational inverses^{} and binary intersections^{}, $\mathcal{B}\subseteq \mathcal{U}$. By construction, each $U\in \mathcal{U}$ contains the element of $U\cap {U}^{1}\in \mathcal{B}$.
There is a useful equivalent^{} condition for being a fundamental system of entourages. Let $\mathcal{B}$ be a nonempty family of subsets of $X\times X$. Then $\mathcal{B}$ is a fundamental system of entourages of a uniformity on $X$ if and only if it the following axioms.

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(B1) If $S$, $T\in \mathcal{B}$, then $S\cap T$ contains an element of $\mathcal{B}$.

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(B2) Each element of $\mathcal{B}$ contains the diagonal $\mathrm{\Delta}(X)$.

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(B3) For any $S\in \mathcal{B}$, the inverse relation of $S$ contains an element of $\mathcal{B}$.

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(B4) For any $S\in \mathcal{B}$, there is an element $T\in \mathcal{B}$ such that the relational composition^{} $T\circ T$ is contained in $S$.
Suppose $\mathcal{B}$ is a fundamental system of entourages for uniformities $\mathcal{U}$ and $\mathcal{V}$. Then $\mathcal{U}\subset \mathcal{V}$. To see this, suppose $S\in \mathcal{U}$. Since $\mathcal{B}$ is a fundamental system of entourages for $\mathcal{U}$, there is some element $B\in \mathcal{B}$ such that $B\subset S$. But $\mathcal{B}\subset \mathcal{V}$, so $B\in \mathcal{V}$. Hence by applying the fact that $\mathcal{V}$ is closed under taking supersets we may conclude that $S\in \mathcal{V}$. So if $\mathcal{B}$ is a fundamental system of entourages, it is a fundamental system for a unique uniformity $\mathcal{U}$. Thus it makes sense to call $\mathcal{U}$ the uniformity generated by the fundamental system $\mathcal{B}$.
References
 1 Nicolas Bourbaki, Elements of Mathematics: General Topology: Part 1, Hermann, 1966.
Title  fundamental system of entourages 

Canonical name  FundamentalSystemOfEntourages 
Date of creation  20130322 16:29:55 
Last modified on  20130322 16:29:55 
Owner  mps (409) 
Last modified by  mps (409) 
Numerical id  5 
Author  mps (409) 
Entry type  Definition 
Classification  msc 54E15 
Defines  uniformity generated by 