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The covering property for σ-ideals of compact sets is an abstract version of the classical perfect set theorem for analytic sets. We will study its consequences using as a paradigm the σ-ideal of countable closed subsets of $2^{\omega }$.

We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.

Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^X$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_{\omega }$ is $F_{\sigma \delta }$. In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_{\omega }$. We show that $S_{\omega }$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover,...

A topological space $X$ is said to be if for all $A\subseteq X$ and all $x\in \overline{A}$ there is $E\subseteq A$ in $ℐ$ such that $x\in \overline{E}$, and is said to be $ℐ$ if whenever a subset $A$ of $X$ contains $\overline{E}$ for every $E\subseteq A$ with $E\in ℐ$, then $A$ itself is closed. An important class of examples are the so called weakly discretely generated spaces (which include sequential, scattered and compact Hausdorff spaces). Another paradigmatic example is the class of Alexandroff spaces which corresponds to spaces generated by finite sets. By considering an appropriate...

We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of $X$ is fixed, we give a necessary and sufficient condition on a point $a\in X^{\text{'}}$ in order that all functions of the Ellis semigroup $E(X,f)$ be continuous at the given point $a$. In the second part, we consider transitive dynamical...