A Universal Continuum of Weight aleph

Topology, Metric Spaces and the Generalized Continuum Hypothesis

This is a paper that aims to interpret the cardinality of a set in terms of Baire Category, i.e. how many closed nowhere dense sets can be deleted from a set before the set itself becomes negligible. . To do this natural tree-theoretic structures such as the Baire topology are introduced, and the Baire Category Theorem is extended to a statement that a $\aleph$-sequentially complete binary tree representation of a Hausdorff topological space that has a clopen base of cardinality $\aleph$ and no isolated or discrete points is not the union of $<\aleph+1$-many nowhere dense subsets for cardinal …

Almost isometric embeddings of metric spaces

We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum separable metric spaces on aleph_1 so that every separable metric space is almost isometrically embedded into one of them when the continuum hypothesis fails. (3) There is no collection of fewer than continuum metric spaces of cardinality aleph_2 so that every ultra-metric space of cardinality aleph_2 is almost isometrically embedded into one of them if aleph_2…

权重为 $\aleph$ 的泛 $P$-群

A universal $P$-group of weight $\aleph$

在连续统假设下，构造了权重至多 ${\mathfrak c}$ 的所有 $P$-群的泛对象，即 $ω$ 的 Čech-Stone 余紧化上同胚群配 $G_δ$-拓扑。

Every metric space of weight $λ=λ^{\aleph_0}$ admits a condensation onto a Banach space

In this paper, we have proved that for each cardinal number $λ$ such that $λ=λ^{\aleph_0}$ a metric space of weight $λ$ admits a bijective continuous mapping onto a Banach space of weight $λ$. Then, we get that every metric space of weight continuum admits a bijective continuous mapping onto the Hilbert cube. This resolves the famous Banach's Problem (when does a metric (possibly Banach) space $X$ admit a bijective continuous mapping onto a compact metric space?) in the class of metric spaces of weight continuum. Also we get that every metric space of weight $λ=λ^{\aleph_0}$ admits a bijective…

Universal flows and automorphisms of $\mathcal P(ω)/\mathrm{fin}$

We prove that for every countable discrete group $G$, there is a $G$-flow on $ω^*$ that has every $G$-flow of weight $\leq\! \aleph_1$ as a quotient. It follows that, under the Continuum Hypothesis, there is a universal $G$-flow of weight $\leq\!\mathfrak{c}$. Applying Stone duality, we deduce that, under \mathsf{CH}, there is a trivial automorphism $τ$ of $\mathcal P(ω)/\mathrm{fin}$ with every other automorphism embedded in it, which means that every other automorphism of $\mathcal P(ω)/\mathrm{fin}$ can be written as the restriction of $τ$ to a suitably chosen subalgebra.

Meager composants of tree-like continua

A subset $M$ of a continuum $X$ is called a \textit{meager composant} if $M$ is maximal with respect to the property that every two of its points are contained in a nowhere dense subcontinuum of $X$. Motivated by questions of Bellamy, Mouron and Ordoñez, we show that no tree-like continuum has a proper open meager composant, and that every tree-like continuum has either $1$ or $2^{\aleph_0}$ meager composants. We also prove a decomposition theorem: If $X$ is tree-like and every indecomposable subcontinuum of $X$ is nowhere dense, then the partition of $X$ into meager composants is upper semi-c…