arXiv subjects
Publications and source records attributed to C. Uzcategui.
The symmetric inverse semigroup $I(X)$ on a set $X$ is the collection of all partial bijections between subsets of $X$ with composition as the algebraic operation. We study a minimal Hausdorff inverse semigroup topologies on $I(X)$. When $X$ is countable, we show some Polish semigroup topologies on $I(X)$.
We study two form of selective selective separability, $SS$ and $SS^+$, on countable spaces with an analytic topology. We show several Ramsey type properties which imply $SS$. For analytic spaces $X$, $SS^+$ is equivalent to have that the collection of dense sets is a $G_δ$ subset of $2^X$, and also equivalent to the existence of a weak base which is an $F_σ$-subset of $2^X$. We study several examples of analytic spaces.
Let $E(X,f)$ be the Ellis semigroup of a dynamical system $(X,f)$ where $X$ is a compact metric space. We analyze the cardinality of $E(X,f)$ for a compact countable metric space $X$. A characterization when $E(X,f)$ and $E(X,f)^* = E(X,f) \setminus \{ f^n : n \in \mathbb{N}\}$ are both finite is given. We show that if the collection of all periods of the periodic points of $(X,f)$ is infinite, then $E(X,f)$ has size $2^{\aleph_0}$. It is also proved that if $(X,f)$ has a point with a dense orbit and all elements of $E(X,f)$ are continuous, then $|E(X,f)| \leq |X|$. For dynamical systems of the form $(ω^2 +1,f)$, we show that if there is a point with a dense orbit, then all elements of $E(ω^2+1,f)$ are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where $E(ω^2+1,f)$ and $ω^2+1$ are homeomorphic but not algebraically homeomorphic, where $ω^2+1$ is taken with the usual ordinal addition as semigroup operation.