On a local property of fences and fans

Abstract: Two closely related classes of topological spaces are fences and fans. A fence is a compact metric space whose components are either arcs or singletons. A fan is a continuum formed by joining arcs at a common vertex, in such a way that intersections of subcontinua are always connected. We prove that every fence can be embedded in the plane and that both fences and fans admit a basis consisting of pierced open sets. This resolves a question by Iztok Bani\v{c}, Goran Erceg, Ivan Jeli\'c, Judy Kennedy, and Van Nall.