The Alexandroff dimension of quotients of \(\mathbb{R}^2\) (Q701785)

A fenestration of \({\mathbb R}^n\) is a collection \({\mathcal W}\) of pairwise disjoint non-empty proper regular open subsets of \({\mathbb R}^n\), whose union is dense in \({\mathbb R}^n\). If \(X\) is a partition of \({\mathbb R}^n\) that contains a fenestration \(\mathcal W\) and the projection map from \({\mathbb R}^n\) to \(X\) (with the quotient topology) is open, then \(X\) is called a \({\mathcal W}\)-grid of \({\mathbb R}^n\). In particular, a \({\mathcal W}\)-grid \(X\) of \({\mathbb R}^n\) is minimal if any continuous open map of \(X\) onto some \({\mathcal W}\)-grid, which is injective on \(\mathcal W\) is a homeomorphism. The authors proved in their previous paper [ibid. 27, 273--286 (2002; Zbl 1053.68108)] that if \(\mathcal W\) is a locally finite fenestration of \({\mathbb R}^n\) whose elements are bounded convex sets, then the Alexandroff dimension of the minimal \(\mathcal W\)-grid is equal to \(n\). Although the question was left open as to whether the result can be generalized to non-convex sets, they gave examples in the same paper to show that it is false for arbitrary locally finite fenestrations of \({\mathbb R}^n\), indeed, the elements of the fenestrations in the examples are non-convex. In this paper, they extend the above result to fenestration of \({\mathbb R}^2\). That is, they show that if \(\mathcal W\) is a locally finite fenestration of \({\mathbb R}^2\) whose elements are homeomorphic to an open 2-disc and whose frontier is homeomorphic to \({\mathbb S}^1\), then the minimal \(\mathcal W\)-grid of \({\mathbb R}^2\) has Alexandroff dimension 2. Furthermore, they give examples to show that the construction of the minimal \(\mathcal W\)-grid in this paper is not applicable to higher dimension.