The topology of digital images

Abstract

In this paper we study those regular fenestrations (as defined by Kronheimer in [3]) that are obtained from a tiling of a topological space. Under weak conditions we obtain that the canonical grid is also the minimal grid associated to each tiling and we prove that it is a T0-Alexandroff semirregular trace space. We also present some examples illustrating how the properties of the grid depend on the properties of the tiling and we pose some questions. Finally we study the topological properties of the grid depending on the properties of the space and the tiling.

Fenestrations induced by perfect tilings

“…This is an interesting point because then the study of the quotient space associated to each grid becomes easier. To this end given a fenestration E of a space X, there is a canonical way to associate a pseudogrid × to E by identifying two points of X if and only if , for every open neighborhood of either, there exists an open neighborhood of the other one which intersects the same collection of elements of E. It is proved in [3] (Theorem 6.2.) that × is the minimal grid associated to E if and only if × is a grid, that is, is a lower semicontinuous decomposition.…”

“…In [3], Kronheimer studied under what conditions the grid is minimal. This is an interesting point because then the study of the quotient space associated to each grid becomes easier.…”

“…According to Kronheimer (see [3]) a fenestration E of a topological space X is a family of pairwise disjoint open sets whose union is dense in X. The fenestration is said to be regular if the open sets of the family are regular open sets.…”

“…Given a fenestration E of a topological space X, a pseudogrid associated to E (see again [3]) is a family of subsets of X such that E ⊂ and is a partition of X. Each pseudogrid determines a quotient space and the quotient map is open if and only if the pseudogrid is lower semicontinuous, that is, St(G, ) = {A ∈ : A ∩ G = ∅} is open for every open set G of X.…”

“…In this paper we study those regular fenestrations (as defined by Kronheimer in [3]) that are obtained from a tiling of a topological space. Under weak conditions we obtain that the canonical grid is also the minimal grid associated to each tiling and we prove that it is a T0-Alexandroff semirregular trace space.…”