This article concerns a class of open billiards consisting of a finite number of strictly convex, non-eclipsing obstacles $K$. The non-wandering set ${{M}_{0}}$ of the billiard ball map is a topological Cantor set, and its Hausdorff dimension has been previously estimated for billiards in ${{\mathbb{R}}^{2}}$ using well-known techniques. We extend these estimates to billiards in ${{\mathbb{R}}^{n}}$ and make various refinements to the estimates. These refinements also allow improvements to other results. We also show that in many cases, the non-wandering set is confined to a particular subset of ${{\mathbb{R}}^{n}}$ formed by the convex hull of points determined by period 2 orbits. This allows more accurate bounds on the constants used in estimating Hausdorff dimension.

Let E be an infinite set, and [E]ω the set of all countably infinite subsets of E. A family ⊂ [E]ω is said to be almost disjoint (respectively, pairwise disjoint) provided for A, B ∈ , if A ≠ B then A ∩ B is finite (respectively, A ∩ B is empty). Moreover, an infinite family A is said to be a maximal almost disjoint family provided it is an infinite almost disjoint family not properly contained in any almost disjoint family. In this paper we are concerned with the following set of topological spaces defined from (maximal) almost disjoint families of infinite subsets of the natural numbers ω.

It is proved that for every Hausdorff space ℝ and for every Hausdorff (regular or Moore) space X, there exists a Hausdorff (regular or Moore, respectively) space S containing X as a closed subspace and having the following properties:

• la) Every continuous map of S into ℝ is constant.

• b) For every point x of S and every open neighbourhood U of x there exists an open neighbourhood V of x, V ⊆ U such that every continuous map of V into ℝ is constant.

• 2) Every continuous map f of S into S (f ≠ identity on S) is constant.