Equivalences (i-MLD) of Tilings and Tiling Spaces

What does it mean when we say that two simple tilings or two simple tiling spaces are equivalent?There are several different notions that explain this, as we will see in this article.By introducing a topology on tiling spaces, there is a notion of a continuous map f:Ω→Ω between two tiling spaces Ω,Ω′.The map f is a homeomorphism if f is 1-1, onto and f -1 is also continuous.For a homeomorphism between simple tiling hulls, we only need to check whether f is continuous, 1-1 and onto, since f -1 is automatically continuous as Ω T is compact.Next, we want to consider continuous maps respectively homeomorphisms, which interact properly with the action of the isometry group on the tiling spaces. Lemma 2.5: Being i-MLD has an equivalence relation on tiling spaces.Proof: Clearly, by definition, i-MLD is reflexive and symmetric.For transitivity, suppose T 1 ,T 2 ∊Ω and 1 ′ ⇒ ′