We give a short proof that if a function is continuous on a countable dense set, then it is continuous on an uncountable set. This is done for functions defined on nonempty complete metric spaces without isolated points, and the argument only uses that Cauchy sequences converge. We discuss how this theorem is a direct consequence of the Baire category theorem, and also discuss Volterra's theorem and the history of this problem. We give a simple example, for each complete metric space without isolated points and each countable subset, of a real-valued function that is discontinuous only on that subset.
A. We study rank-two symbolic systems (as topological dynamical systems) and prove that the Thue-Morse sequence and quadratic Sturmian sequences are rank-two and define rank-two symbolic systems. IRank-one measure-preserving transformations have played an important role in ergodic theory since the pioneering work of Chacón [4]. In this paper, rather than considering the notion of rankone in measurable dynamics we are interested in studying rank-one and higher rank systems strictly from the point of view of topological dynamics, in particular as symbolic shifts. Of course, symbolic systems have been used extensively in ergodic theory. In [13], Kalikow discusses a symbolic model for rank-one measure-preserving transformations, and Ferenczi [8] in his survey on rank-one finite measure-preserving transformations mentions the symbolic definition for rank-one transformations. Later in [3], Bourgain used a class of symbolic rank-one transformations for which he proved the Moebius disjointness law, and rank-one symbolic shifts are also considered in [1,5,7]. It was in [11] that Gao and Hill started a systematic study of (non-degenerate) rank-one shifts as topological dynamical systems, and proved several properties for rank-one symbolic shifts. In this paper we study higher rank systems and prove that the system defined by the Thue-Morse sequence and systems defined by quadratic Sturmian sequences are rank-two.The terminology "rank-one" comes from "rank-one" cutting and stacking systems [4]. As shown by Kalikow [13], one can encode cutting and stacking systems as a shift on a symbolic system and he shows that the two systems are measurably isomorphic when the symbolic sequence is non-periodic. Gao and Hill introduced the notion of (symbolic) rank-one shifts as an extension to earlier ideas and developed various results associated to symbolic rank-one systems. We start by first defining rank-one words using the definition provided by Gao and Hill in [11].Definition 1.1. Let F be the set consisting of all finite words in the alphabet {0, 1} that begin and end with 0. Let V ∈ {0, 1} N . We say that V is built from v ∈ F if there exists a sequence {a i } i≥1 of natural numbers such that V = v1 a 1 v1 a 2 v • • • . LetWe say that the infinite word V is rank-one if A V is infinite.
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