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On the structure of the intersection of two middle third Cantor sets.

Gregory J. Davis, Tian You Hu (1995)

Publicacions Matemàtiques

Motivated by the study of planar homoclinic bifurcations, in this paper we describe how the intersection of two middle third Cantor sets changes as the sets are translated across each other. The resulting description shows that the intersection is never empty; in fact, the intersection can be either finite or infinite in size. We show that when the intersection is finite then the number of points in the intersection will be either 2n or 3 · 2n. We also explore the Hausdorff dimension of the intersection...

On (transfinite) small inductive dimension of products

Vitalij A. Chatyrko, Konstantin L. Kozlov (2000)

Commentationes Mathematicae Universitatis Carolinae

In this paper we study the behavior of the (transfinite) small inductive dimension ( t r i n d ) i n d on finite products of topological spaces. In particular we essentially improve Toulmin’s estimation [T] of t r i n d for Cartesian products.

On uncountable collections of continua and their span

Dušan Repovš, Arkadij Skopenkov, Evgenij Ščepin (1996)

Colloquium Mathematicae

We prove that if the Euclidean plane ℝ 2 contains an uncountable collection of pairwise disjoint copies of a tree-like continuum X, then the symmetric span of X is zero, sX = 0. We also construct a modification of the Oversteegen-Tymchatyn example: for each ε > 0 there exists a tree X ⊂ ℝ 2 such that σX < ε but X cannot be covered by any 1-chain. These are partial solutions of some well-known problems in continua theory.

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