Factoring Continuous Homomorphisms Defined on Submonoids of Products of Topologized Monoids

We study factorization properties of continuous homomorphisms defined on submonoids of products of topologized monoids. We prove that if S is an ω-retractable submonoid of a product D = ∏ i ∈ I D i of topologized monoids and f : S → H is a continuous homomorphism to a topologized semigroup H with ψ ( H ) ≤ ω , then one can find a countable subset E of I and a continuous homomorphism g : p E ( S ) → H satisfying f = g ∘ p E ↾ S , where p E is the projection of D to ∏ i ∈ E D i . The same conclusion is valid if S contains the Σ -product Σ D ⊂ D . Furthermore, we show that in both cases, there exists the smallest by inclusion subset E ⊂ I with the aforementioned properties.

A note on discrete C-embedded subspaces

It is shown that in some non-discrete topological spaces, discrete subspaces with certain cardinality are C-embedded. In particular, this generalizes the well-known fact that every countable subset of P-spaces are C-embedded. In the presence of the measurable cardinals, we observe that if X is a discrete space then every subspace of υ X (i.e., the Hewitt realcompactification of X) whose cardinal is nonmeasurable, is a C-embedded, discrete realcompact subspace of υ X. This generalizes the well-known fact that the discrete spaces with nonmeasurable cardinal are realcompact.

NAMBA FORCING, WEAK APPROXIMATION, AND GUESSING

We prove a variation of Easton’s lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle IGMP, GMP together with 2ω ≤ ω2 is consistent with the existence of an ω1-distributive nowhere c.c.c. forcing poset of size ω1. We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle GMP follow from the existence of stationarily many weakly guessing models. Using Namba forcing, we construct a model in which there are stationarily many indestructibly weakly guessing models which have a bounded countable subset not covered by any countable set in the model.

On Annihilator Ideals of Pseudo-differential Operator Rings

Let R be a ring with a derivation δ and R((x-1; δ)) denote the pseudo-differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R((x-1; δ)). Among applications, it is shown that for an Armendariz ring R of pseudo-differential operator type, the ring R((x-1; δ)) is Baer (resp., quasi-Baer, PP, right zip) if and only if R is a Baer (resp., quasi-Baer, PP, right zip) ring. For a δ-weakly rigid ring R, R((x-1; δ)) is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.

Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

We show that for a coanalytic subspace X of 2ω, the countable dense homogeneity of Xω is equivalent to X being Polish. This strengthens a result of Hruˇs´ak and Zamora Avilés. Then, inspired by results of Hernández-Guti´errez, Hruˇs´ak, and van Mill, using a technique of Medvedev, we construct a non-Polish subspace X of 2ω such that Xω is countable dense homogeneous. This gives the ûrst ZFC answer to a question of Hruˇs´ak and Zamora Avil´es. Furthermore, since our example is consistently analytic, the equivalence result mentioned above is sharp. Our results also answer a question of Medini and Milovich. Finally, we show that if every countable subset of a zero-dimensional separable metrizable space X is included in a Polish subspace of X, then Xω is countable dense homogeneous.

Separability of Real Normed Spaces and Its Basic Properties

Summary In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the real numbers. In the second section, the basic properties of the separable normed spaces are discussed. It is applied to isomorphic spaces via bounded linear operators and double dual spaces. In the last section, it is proved that the completeness and reflexivity are transferred to sublinear normed spaces. The formalization is based on [34], and also referred to [7], [14] and [16].

Some remarks on almost star countable spaces

A space X is almost star countable (weakly star countable) if for each open cover U of X there exists a countable subset F of X such that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\bigcup {_{x \in F}\overline {St\left( {x,U} \right)} } = X$ \end{document} (respectively, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\overline {\bigcup {_{x \in F}} St\left( {x,U} \right)} = X$ \end{document}. In this paper, we investigate the relationships among star countable spaces, almost star countable spaces and weakly star countable spaces, and also study topological properties of almost star countable spaces.

Schmidt games and non-dense forward orbits of certain partially hyperbolic systems

Let$f:M\rightarrow M$be a partially hyperbolic diffeomorphism with conformality on unstable manifolds. Consider a set of points with non-dense forward orbit:$E(f,y):=\{z\in M:y\notin \overline{\{f^{k}(z),k\in \mathbb{N}\}}\}$for some$y\in M$. Define$E_{x}(f,y):=E(f,y)\cap W^{u}(x)$for any$x\in M$. Following a method of Broderick, Fishman and Kleinbock [Schmidt’s game, fractals, and orbits of toral endomorphisms.Ergod. Th. & Dynam. Sys.31(2011), 1095–1107], we show that$E_{x}(f,y)$is a winning set for Schmidt games played on$W^{u}(x)$which implies that$E_{x}(f,y)$has Hausdorff dimension equal to$\dim W^{u}(x)$. Furthermore, we show that for any non-empty open set$V\subset M$,$E(f,y)\cap V$has full Hausdorff dimension equal to$\dim M$, by constructing measures supported on$E(f,y)\cap V$with lower pointwise dimension converging to$\dim M$and with conditional measures supported on$E_{x}(f,y)\cap V$. The results can be extended to the set of points with forward orbit staying away from a countable subset of$M$.

On principally quasi-Baer skew power series rings

Let [Formula: see text] be a monomorphism of a ring [Formula: see text] which is not assumed to be surjective. It is shown that, for an [Formula: see text]-weakly rigid [Formula: see text], the skew power series ring [Formula: see text] is right p.q.-Baer if and only if the skew Laurent series ring [Formula: see text] is right p.q.-Baer if and only if [Formula: see text] is right p.q.-Baer and every countable subset of right semicentral idempotents has a generalized countable join.

Infinite partition monoids

Let 𝒫X and 𝒮X be the partition monoid and symmetric group on an infinite set X. We show that 𝒫X may be generated by 𝒮X together with two (but no fewer) additional partitions, and we classify the pairs α, β ∈ 𝒫X for which 𝒫X is generated by 𝒮X ∪ {α, β}. We also show that 𝒫X may be generated by the set ℰX of all idempotent partitions together with two (but no fewer) additional partitions. In fact, 𝒫X is generated by ℰX ∪ {α, β} if and only if it is generated by ℰX ∪ 𝒮X ∪ {α, β}. We also classify the pairs α, β ∈ 𝒫X for which 𝒫X is generated by ℰX ∪ {α, β}. Among other results, we show that any countable subset of 𝒫X is contained in a 4-generated subsemigroup of 𝒫X, and that the length function on 𝒫X is bounded with respect to any generating set.