Abstract. 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.

This paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space X which are obtained by deleting singletons determine X uniquely up to homeomorphism. If the question can be answered affirmatively, such a space is called reconstructible.We prove that in various cases topological properties can be reconstructed. As main result we find that familiar spaces such as the reals R, the rationals Q and the irrationals P are reconstructible, as well as spaces occurring as Stone-Čech compactifications. Moreover, some non-reconstructible spaces are discovered, amongst them the Cantor set C.

Abstract. 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.1. Introduction. Given a cardinal κ, a subset of the plane is said to be a κ-point set if and only if it meets every line in exactly κ many points and is said to be a partial κ-point set if and only if it meets every line in at most κ many points. We are particularly interested in the case that κ = 2, and we refer to such sets as two-point sets.The existence of two-point sets was first shown by Mazurkiewicz [8]. (A French translation is given in [9].) A two-point set is classically constructed via transfinite recursion, making use of an arbitrary well-ordering of the collection of all lines in the plane. We will demonstrate two constructions of two-point sets which result by varying the use of this well-ordering.The first construction shows that working in ZFC+CH, we can construct a two-point set which is contained in the union of a countable collection of concentric circles of unbounded radius. In general, classical constructions of two-point sets can be thought of as starting with a c-point set and then refining it to a two-point set, and so this construction is of interest because we obtain a two-point set by refining an ℵ 0 -point set. We achieve this development by being careful about the well-ordering of the collection of all lines that we choose to help us in our construction.The second construction shows that in ZF, if we assume that the real line satisfies a certain number of properties, we can construct a two-point set without explicitly invoking the Axiom of Choice. It is a question of Erdős,

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A two-point set is a subset of the plane which meets every line in exactly two points. We discuss previous work on the topological symmetries of a two-point set, and show that there exist subgroups of S 1 which do not leave any two-point set invariant. Further, we show that two-point sets may be chosen to be topological groups, in which case they are also homogeneous.