Uniformity iff Quasiuniformity has Symmetric Basis

Theorem

Let $S$ be a set.

Let $\UU$ be a quasiuniformity on $S$.


Then $\UU$ is a uniformity if and only if $\UU$ has a symmetric filter basis.


Proof

Let $\UU$ be a quasiuniformity on $S$ which has a symmetric filter basis $\BB$.

From the definition of filter basis, all the elements of $\UU$ can be formed from intersections of elements of $\BB$.

But from Intersection of Symmetric Relations is Symmetric, it follows that all elements of $\UU$ are symmetric.


Now suppose $\UU$ is a uniformity.

If $\BB$ is a filter basis of $\UU$ then all the elements of $\BB$ are also elements of $\UU$.

Hence $\BB$ is a symmetric filter basis of $\UU$.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Uniformities
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