Abbott Dimension, Mathematics Inspired by Flatland

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.Comment: V3: Made edits for clarity and ease of reading. Added examples to later section