Cauchy conditions on symmetrics

Abstract

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