Abbott Dimension, Mathematics Inspired by Flatland

Abstract

In this paper we introduce the Abbott dimension of Hausdorff spaces, an intuitively defined dimension function inspired by Edwin Abbott's \emph{Flatland}. We show that on separable metric spaces the Abbott dimension is bounded above by the large inductive dimension. Consequently we show that the Abbott dimension of \(\mathbb{R}^{n}\) is \(n\). We conclude by showing that hereditarily indecomposable continua all have Abbott dimension \(1\), while there exist such continua of arbitrarily high large inductive, small inductive, and covering dimension.