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Fractal sets satisfying the strong open set condition in complete metric spaces

Goodman G. S.

Opuscula Mathematica

2008

Vol. 28, no. 4

463-470

Let K be a Hutchinson fractal in a complete metric space X, invariant under the action S of the union of a finite number of Lipschitz contractions. The Open Set Condition states that X has a non-empty subinvariant bounded open subset V, whose images under the maps are disjoint. It is said to be strong if V meets K. We show by a category argument that when K ⊄ V and the restrictions of the contractions to V are open, the strong condition implies that [formula] termed the core of V, is non-empty. In this case, it is an invariant, proper, dense, subset of K, made up of points whose addresses are unique. Conversely, [formula] implies the SOSC, without any openness assumption.

Finite partitions of topological groups into congruent thick subsets

Weber H., Zoli E.

Journal of Applied Analysis

2008

Vol. 14, nr 1

1-12

We establish fairly general sufficient conditions for a locally compact group (a Baire topological group) to admit partitions into finitely many congruent μ-thick (everywhere of second category) subset.

On non-injectivity of Borel functions selecting points from compact sets. 2

Balcerzak M., Peredko J., Pol R.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 2

103-105

The note is a supplement to (1]. We refine a result from [1] on the non-injectivity of Borel selections on the hyperspaces and we discuss the relations of the results in [1] with some results obtained by Lecomte (4].

A result related to the problem cn of Fremlin

Kolman O., Shelah S.

Journal of Applied Analysis

1998

Vol. 4, nr 2

161-165

We show that the set of injective functions from any uncountable cardinal less than the continuum into the real numbers is of the second category in the box product topology.