TL;DR
This paper investigates properties of fences and fans, two classes of topological spaces, proving their embeddability in the plane and the existence of bases of pierced open sets, thus answering a specific open question.
Contribution
It establishes that all fences can be embedded in the plane and both fences and fans have bases of pierced open sets, resolving an open problem in topology.
Findings
Fences are embeddable in the plane.
Both fences and fans admit bases of pierced open sets.
The results answer an open question by several topologists.
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.
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