# Two definitions of totally bounded uniform spaces

Wikipedia gives this definition of totally bounded uniform space:

a subset $S$ of a uniform space $X$ is totally bounded if and only if, given any entourage $E$ in $X$, there exists a finite cover of $S$ by subsets of $X$ each of whose Cartesian squares is a subset of $E$.

Minimus Heximus user of math.stackexchange.com has given in his answer an other definition:

$(X,\mathcal D)$ is totally bounded, when for each entourage $D\in \mathcal D$, there are $x_1,...,x_n\in X$ with $D[x_1]\cup...\cup D[x_n]=X$.

Minimus Heximus has proved that his definition follows from Wikipedia's definition.

Does the converse hold? Is the Wikipedia's definition a consequence of Minimus Heximus's definition? Or is there a counter-example?

Given any entourage $E$, let $\:U=E\:$,$\:$ and let $V$ be as in part 4 of
wikipedia's definition of uniform space, and let $\;\; D \: = \: V^{-1} \cap V \;\;\;$.

$D \;\; = \;\; V^{-1} \cap V \;\; \subseteq \;\; V^{-1} \;\;\;\;$ and $\;\;\;\; D \;\; = \;\; V^{-1} \cap V \;\; \subseteq \;\; V \;\;\;\;$ and $\;\;\;\;$ $D$ is an entourage

Let $\:x_1,...,x_n\:$ be such that $\;\; D[x_1\hspace{-0.03 in}]\cup...\cup D[x_n\hspace{-0.03 in}] \: = \: X \;\;\;$.

For all elements $i$ of $\:\{\hspace{-0.02 in}1,\hspace{-0.02 in}2,\hspace{-0.02 in}3,...,n\}\:$,$\:$ for all elements $\:\langle \hspace{.03 in}y,\hspace{-0.02 in}z\rangle\:$ of $X^{\hspace{.02 in}2}$,

$\langle \hspace{.03 in}y,\hspace{-0.02 in}z\rangle \: \in \: \left(D[\hspace{.02 in}x_i]\right)^2 \;\;\;\; \implies \;\;\;\; \langle \hspace{.02 in}x_i,y\rangle \: \in \: D \: \subseteq \: V^{-1} \;\; \text{ and } \;\; \langle \hspace{.02 in}x_i,z\rangle \: \in \: D \: \subseteq \: V$
$\implies \;\;\;\; \langle \hspace{.03 in}y,x_i\rangle \: \in \: V \;\; \text{ and } \;\; \langle \hspace{.02 in}x_i,z\rangle \: \in \: V \;\;\;\; \implies \;\;\;\; \langle \hspace{.03 in}y,z\rangle \: \in \: U \: = \: E$

.

For all elements $i$ of $\:\{\hspace{-0.02 in}1,\hspace{-0.02 in}2,\hspace{-0.02 in}3,...,n\}\:$,$\:$ $\: \left(D[\hspace{.02 in}x_i]\right)^2 \subseteq E \:\:$.

$D[x_1\hspace{-0.03 in}]\hspace{.02 in},...,D[x_n\hspace{-0.03 in}]\:$ satisfies wikipedia's definition.

Since that works for any entourage $E$, Wikipedia's definition follows from his definition.

Therefore Wikipedia's definition is a consequence of Minimus Heximus's definition.