Tilings in topological spaces

Abstract

Abstract. A tiling of a topological space X is a covering of X by sets (called tiles) which are the closures of their pairwise-disjoint interiors. Tilings of R 2 have received considerable attention (see [2] for a wealth of interesting examples and results as well as an extensive bibliography). On the other hand, the study of tilings of general topological spaces is just beginning (see [1,3,4,6]). We give some generalizations for topological spaces of some results known for certain classes of tilings of topological vector spaces.

Hahn--Mazurkiewicz revisited: a generalization

“…Let Γ be a covering of X. Γ is said to be locally finite if for all x ∈ X there exists a neighborhood of x which meets only a finite number of elements of Γ. Γ is said to be a tiling, if all elements of Γ are regularly closed (a subset is regularly closed ([29, Problem 3.D]) if it is the closure of its interior) and they have disjoint interiors (see [1]). Definition 2.1.…”

Fenestrations induced by perfect tilings

“…On the other hand, the study of tilings of general topological spaces is just beginning (see [1], [4], [5] and [6]). …”