Let ε > 0 and F be a family of finite subsets of the Cantor set C. Following D. H. Fremlin, we say that F is ε-filling over C if F is hereditary and for every F ⊆ C finite there exists G ⊆ F such that G ∈ F and |G| ≥ ε|F |. We show that if F is ε-filling over C and C-measurable in [C], then for every P ⊆ C perfect there exists Q ⊆ P perfect with [Q] ⊆ F . A similar result for weaker versions of...

Whereas Cantor multilayers made of an isotropic dielectric–magnetic material with positive refractive index will show power–law characteristics, low–order Cantor multilayers made of materials with negative refractive index will not exhibit the power– law nature. A reason for this anomalous behavior is presented.

We describe the Ziegler spectrum of a Bézout domain B = D + XQ[X] where D is a principal ideal domain and Q is its field of fractions; in particular we compute the Cantor–Bendixson rank of this space. Using this, we prove the decidability of the theory of B-modules when D is “sufficiently” recursive.

Cantor’s proof of this result [2, §2] makes use of nested intervals, but today a proof based on another ingenious idea of Cantor is more popular, namely the diagonal method, which he introduced in 1891 to prove the uncountability of 2 [3]. However, Cantor himself did not employ diagonalization directly in the proof of uncountability of R, but gave a rather intricate derivation of ∣ ∣2 ∣ ∣ = |R|...

We describe all possible ways of bi-ordering Thompson group F: its space of bi-orderings is made up of eight isolated points and four canonical copies of the Cantor set.

We describe and characterize all homogeneous subsets of the Cantor set which are both an F„s and a GSo; it turns out that there are wx such spaces.

We show that, for any n ≠ 2, most orientation preserving homeomorphisms of the sphere S2n have a Cantor set of fixed points. In other words, the set of such homeomorphisms that do not have a Cantor set of fixed points is of the first Baire category within the set of all homeomorphisms. Similarly, most orientation reversing homeomorphisms of the sphere S2n+1 have a Cantor set of fixed points for...