totally bounded uniform space

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Defines:
totally bounded, totally bounded uniformity
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:

Mathematics Subject Classification

An alternate definition of totally bounded uniform space 2

Sorry, my wrong LaTeX. I repeat the comment again.

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 PlanetMath definition.

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