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Set-theoretic constructions of two-point sets

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Chad B., Knight R., Suabedissen R.

Fundamenta Mathematicae

2009

tom 203

nr 2

179-189

A two-point set is a subset of the plane which meets every line in exactly two points. By working in models of set theory other than ZFC, we demonstrate two new constructions of two-point sets. Our first construction shows that in ZFC + CH there exist two-point sets which are contained within the union of a countable collection of concentric circles. Our second construction shows that in certain models of ZF, we can show the existence of two-point sets without explicitly invoking the Axiom of Choice.

Uncountable ω-limit sets with isolated points

100%

Good C., Raines B., Suabedissen R.

Fundamenta Mathematicae

2009

tom 205

nr 2

179-189

We give two examples of tent maps with uncountable (as it happens, post-critical) ω-limit sets, which have isolated points, with interesting structures. Such ω-limit sets must be of the form C ∪ R, where C is a Cantor set and R is a scattered set. Firstly, it is known that there is a restriction on the topological structure of countable ω-limit sets for finite-to-one maps satisfying at least some weak form of expansivity. We show that this restriction does not hold if the ω-limit set is uncountable. Secondly, we give an example of an ω-limit set of the form C ∪ R for which the Cantor set C is minimal.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Suabedissen R.

1 Chad B.

1 Good C.

1 Knight R.

1 Raines B.

2 2009

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Set-theoretic constructions of two-point sets

Uncountable ω-limit sets with isolated points