On a local property of fences and fans

David S. Lipham

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arXiv · 2508.03530

On a local property of fences and fans

David S. Lipham

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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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David S. Lipham. 2025-08-05. On a local property of fences and fans. https://arxiv.org/abs/2508.03530

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