Abbott Dimension, Mathematics Inspired by Flatland

Abstract

AbstractWhat is the “right way” to define dimension? Mathematicians working in the early and middle 20th-century formalized three intuitive definitions of dimension that all turned out to be equivalent on separable metric spaces. But were these definitions the “right” ones? What would it mean to have the “right” definition of dimension? In this paper we attempt to inspire thought about these questions by introducing Abbott dimension, a geometrically intuitive definition of dimension based on Edwin Abbott’s 1884 novella Flatland. We show that while Abbott dimension has intuitive appeal, it does not always agree with the classical definitions of dimension on separable metric spaces.MSC: 54F45 AcknowledgmentThe author would like to thank Kate Franklin for her reading of, and input on, early drafts of this paper. The author would also like to thank the referees and editors for this paper whose input greatly improved it. This work was supported by the Israel Science Foundation grant No. 2196/20Notes1 Alternative formulations of what a separator could be do exist. Indeed, in what was perhaps the first precise formulation of dimension, Brouwer in [Citation5] defined his invariant, the dimensiongrad, of a space. This invariant is identical in definition to the large inductive dimension we have defined with the difference being that instead of using separators it uses the notion of a cut. A cut in a space X between disjoint closed subsets A and B is a third closed set C⊆X that is disjoint from A and B and is such that any continuum K⊆X that intersects A and B must also intersect C. This definition of dimension agrees with the classical definitions mentioned in the introduction on the class of compact metrizable spaces (see [Citation7]).2 Interestingly, for the classical definitions of dimension the result does not hold if the space is infinite dimensional. In [Citation8] a compact metric space of infinite dimension (with respect to any of the classical definitions) is constructed such that every subspace is either also infinite dimensional, or zero dimensional.3 In 1920 Knaster and Kuratowski asked in [Citation12] if a nondegenerate homogeneous continuum in the plane must be a closed curve. The next year Mazurkiewicz asked in [Citation13] if every continuum in the plane which is homeomorphic to all of its nondegenerate subcontinua must be an arc. In [Citation11] Knaster would give an example of a hereditarily indecomposable continuum in 1923. In [Citation2] Bing would answer Knaster and Kuratowski’s question negatively with his own construction in 1948, and Moise would answer Mazurkiewicz’s question negatively in [Citation14]. Moise dubbed his example the “pseudoarc” due to its similarity to an arc. Bing would go on in [Citation3] to show that his, Knaster’s, and Moise’s examples were all homeomorphic. This history and the results surrounding the pseudoarc can be found in [Citation15].4 One may denote the set of all continua in Rn by C(Rn) and endow it with the Hausdorff metric dh. Given two continua X,Y∈C(Rn),dh(X,Y) is the infimum over all ϵ>0 such that X⊆B(Y,ϵ) and Y⊆B(X,ϵ). Here B(X,ϵ) is the open metric ball about X of radius ϵ. The result we reference says that hereditarily indecomposable continua are a dense subset of this space.Additional informationNotes on contributorsJeremy SiegertJeremy Siegert received his Ph.D. in mathematics from the University of Tennessee Knoxville. He is currently a post-doctoral fellow at the Ben Gurion University of the Negev in Beer Sheva, Israel.