Cauchy conditions on symmetrics

Abstract:

We call a symmetric $d$ on a space $X$ a ${\text {wC}}$ symmetric if whenever $A \subseteq X$ and there exists $\varepsilon > 0$ such that $d(x,y) \geqslant \varepsilon$ for all $x$, $y \in A$, then $A$ is relatively discrete. We show that there are no $L$-spaces which admit ${\text {wC}}$ symmetries. The ${\text {wC}}$ notion is extended to certain weaker structures such as $\mathcal {F}$-spaces with similar results.