Geometrical tilings : distance, topology, compactness and completeness

TL;DR

This paper reviews various metrics on tilings of Euclidean space, proves their correctness, explores their topological and metric equivalences for FLC subshifts, and provides complete proofs of classical results on compactness and completeness.

Contribution

It clarifies and proves the properties of different tiling metrics, including their equivalences and classical topological results, with complete proofs where lacking.

Findings

Most tiling metrics are valid and topologically equivalent for FLC subshifts.

Classical results on compactness and completeness are reaffirmed with full proofs.

The paper consolidates and clarifies foundational results on tiling metrics.

Abstract

We present the different distances on tilings of Rd that exist in the literature, we prove that (most of) these definitions are correct (i.e. they indeed define metrics on tilings of Rd ). We prove that for subshifts with finite local complexity (FLC) these metrics are topologically equivalent and even metrically equivalent, and also we present classical results of compactness and completeness. Note that, excluding the equivalence of these metrics, all of the results presented here are known (see for example the survey [Rob04]) however we were unable to find a reference with complete proofs for some of these results so we decided to write this notice to clarify some definitions and give full proofs.