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Register a new userIterates of dynamical systems on compact metrizable countable spaces
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Given a dynamical system $(X,f)$, we let $E(X,f)$ denote its Ellis semigroup and $E(X,f)^* = E(X,f) setminus {f^n : n in mathbb{N}}$. We analyze the Ellis semigroup of a dynamical system having a compact metric countable space as a phase space. We show that if $(X,f)$ is a dynamical system such that $X$ is a compact metric countable space and every accumulation point $X$ is periodic, then either each function of $E(X,f)^*$ is continuous or each function of $E(X,f)^*$ is discontinuous. We describe an example of a dynamical system $(X,f)$ where $X$ is a compact metric countable space, the orbit of each accumulation point is finite and $E(X,f)^*$ contains continuous and discontinuous functions.
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We prove that each metrizable space (of cardinality less or equal to continuum) has a (first countable) uniform Eberlein compactification and each scattered metrizable space has a scattered hereditarily paracompact compactification. Each compact scattered hereditarily paracompact space is uniform Eberlein and belongs to the smallest class of compact spaces, that contain the empty set, the singleton, and is closed under producing the Aleksandrov compactification of the topological sum of a family of compacta from that class.
Given a Lie group $G$ we study the class $M$ of proper metrizable $G$-spaces with metrizable orbit spaces, and show that any $G$-space $X in M$ admits a closed $G$-embedding into a convex $G$-subset $C$ of some locally convex linear $G$-space, such that $X$ has some $G$-neighborhood in $C$ which belongs to the class $M$. As corollaries we see that any $G$-ANE for $M$ has the $G$-homotopy type of some $G$-CW complex and that any $G$-ANR for $M$ is a $G$-ANE for $M$.
A theorem by Norman L. Noble from 1970 asserts that every product of completely regular, locally pseudo-compact k_R-spaces is a k_R-space. As a consequence, all direct products of locally compact Hausdorff spaces are k_R-spaces. We provide a streamlined proof for this fact.
We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of omega_1 is hereditarily paracompact.
A Hausdorff topological space $X$ is called $textit{superconnected}$ (resp. $textit{coregular}$) if for any nonempty open sets $U_1,dots U_nsubseteq X$, the intersection of their closures $bar U_1capdotscapbar U_n$ is not empty (resp. the complement $Xsetminus (bar U_1capdotscapbar U_n)$ is a regular topological space). A canonical example of a coregular superconnected space is the projective space $mathbb Qmathsf P^infty$ of the topological vector space $mathbb Q^{<omega}={(x_n)_{ninomega}in mathbb Q^{omega}:|{ninomega:x_n e 0}|<omega}$ over the field of rationals $mathbb Q$. The space $mathbb Qmathsf P^infty$ is the quotient space of $mathbb Q^{<omega}setminus{0}^omega$ by the equivalence relation $xsim y$ iff $mathbb Q{cdot}x=mathbb Q{cdot}y$. We prove that every countable second-countable coregular space is homeomorphic to a subspace of $mathbb Qmathsf P^infty$, and a topological space $X$ is homeomorphic to $mathbb Qmathsf P^infty$ if and only if $X$ is countable, second-countable, and admits a decreasing sequence of closed sets $(X_n)_{ninomega}$ such that (i) $X_0=X$, $bigcap_{ninomega}X_n=emptyset$, (ii) for every $ninomega$ and a nonempty open set $Usubseteq X_n$ the closure $bar U$ contains some set $X_m$, and (iii) for every $ninomega$ the complement $Xsetminus X_n$ is a regular topological space. Using this topological characterization of $mathbb Qmathsf P^infty$ we find topological copies of the space $mathbb Qmathsf P^infty$ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.
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