The author provides a definition of dimension for topological spaces inspired by notions that appear in the satirical novella, Flatland, by Edwin Abbott that appeared in 1884. Alongside Abbott’s satire of society, there are serious mathematical and geometrical concepts afloat. In particular, he tells a story about residents of a world that consists only of a plane (Flatland) and how they would have to exist in such a realm. A visitor from 3-space would be able to see into them as would a visitor from 4-space be able to see into the denizens of 3-space. So the idea of Siegert is to devise a definition of dimension that would be based on “seeing into.” This is the foundation of what is called the “Abbott dimension,” \(\mathrm{Ab}(X)\), for any nonempty Hausdorff space \(X\) (Definition 11.).
In this paper it is shown that \(\mathrm{Ab}(\mathbb{R}^n)=n\), which is a fundamental requirement of any theory of topological dimension. On the other hand, the author provides a host of examples that show that the dimension \(\mathrm{Ab}\) does not always equal the classical dimension \(\mathrm{dim}\) for separable metrizable spaces even when they are compact (examples of the latter involve hereditarily indecomposable continua).