Answering a question on relative countable paracompactness

Countable dense homogeneity and $λ$-sets

We show that all sufficiently nice $λ$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $κ\le \mathfrak{b}$ there is a countable dense homogeneous metric space of size $κ$. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size $κ$ is equivalent to the existence of a $λ$-set of size $κ$. On the other hand, it is consistent with the continuum arbitrarily large that every $\mathsf{CDH}$ metric space has size either $ω_1$ or size $\mathfrak c$. An example of a Baire $\mathsf{CDH}$ metric s…

On first countable, cellular-compact spaces

As it was introduced by Tkachuk and Wilson, a topological space $X$ is cellular-compact if given any cellular, i.e. disjoint, family $\mathcal U$ of non-empty open subsets of $X$ there is a compact subspace $K\subset X$ such that $K\cap U\ne \emptyset$ for each $U\in \mathcal U$. Answering several questions raised by Tkachuk and Wilson we show that (1) any first countable cellular-compact $T_2$ space is $T_3$, and so its cardinality is at most $\mathfrak{c} = 2^ω$; (2) $cov(\mathcal M)>ω_1$ implies that every first countable and separable cellular-compact $T_2$ space is compact; (3 if there is…

On some classes of Lindelöf Sigma-spaces

We consider special subclasses of the class of Lindelöf Sigma-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space $X$ is in the class $LΣ(\leqκ)$ if it admits a cover by compact subspaces of weight $κ$ and a countable network for the cover. We restrict our attention to $κ\leqω$. In the case $κ=ω$, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the $P$-approximable spaces considered by Tkacenko. The case $κ=1$ corresponds to the spaces of countable network weight, but even the case …

On the submetrizability number and $i$-weight of quasi-uniform spaces and paratopological groups

We derive many upper bounds on the submetrizability number and $i$-weight of paratopological groups and topological monoids with open shifts. In particular, we prove that each first countable Hausdorff paratopological group is submetrizable thus answering a problem of Arhangelskii posed in 2002. Also we construct an example of a zero-dimensional (and hence regular) Hausdorff paratopological abelian group $G$ with countable pseudocharacter which is not submetrizable. In fact, all results on the $i$-weight and submetrizability are derived from more general results concerning normally quasi-unifo…

Submaximal spaces and cardinal invariants

We give a combinatorial characterization of countable submaximal subspaces of $2^κ$. Using a parametrized version of Mathias forcing, we prove that there exists a countable submaximal subspace of $2^{ω_1}$ whilst $\mathfrak{c}=ω_2$. Combining this with previous results, we construct a disjointly tight countable irresolvable space of weight $<\mathfrak{c}$, answering a question of Bella and Hrušák.

类 $C(ω_1)$ 与可数网络权

The class $C(ω_1)$ and countable net weight

本文研究拓扑空间类 $C(ω_1)$ 与可数网络权之间的关系，通过独立性结果证明连续统假设及更弱的 super stick 原理可推出 $C(ω_1)$ 中正则空间具有可数网络权，回答了 Hart 和 Kunen 提出的问题。

回答可分解半拓扑群中Tkachenko的若干问题

Answer some Tkachenko's questions in factorizable semitopological groups

本文构造了一个满足特定拓扑性质的拟拓扑阿贝尔群，对Tkachenko提出的多个半拓扑群相关问题给出否定或肯定回答，并证明可数半拓扑群的可数族可通过第二可数Tychonoff半拓扑群分解。

First countability, $ω$-well-filtered spaces and reflections

We first introduce and study two new classes of subsets in $T_0$ spaces - $ω$-Rudin sets and $ω$-well-filtered determined sets lying between the class of all closures of countable directed subsets and that of irreducible closed subsets, and two new types of spaces - $ω$-$d$ spaces and $ω$-well-filtered spaces. We prove that an $ω$-well-filtered $T_0$ space is locally compact iff it is core compact. One immediate corollary is that every core compact well-filtered space is sober, answering Jia-Jung problem with a new method. We also prove that all irreducible closed subsets in a first countable …

A Note on Monotonically Metacompact Spaces

We show that any metacompact Moore space is monotonically metacompact and use that result to characterize monotone metacompactness in certain generalized ordered (GO)spaces. We show, for example, that a generalized ordered space with a sigma-closed-discrete dense subset is metrizable if and only if it is monotonically (countably) metacompact, that a monotonically (countably) metacompact GO-space is hereditarily paracompact, and that a locally countably compact GO-space is metrizable if and only if it is monotonically (countably) metacompact. We give an example of a non-metrizable LOTS that is …

Pointwise convergence of partial functions: The Gerlits-Nagy Problem

For a set $X\sbst\R$, let $B(X)\sbst\R^X$ denote the space of Borel real-valued functions on $X$, with the topology inherited from the Tychonoff product $\R^X$. Assume that for each countable $A\sbst B(X)$, each $f$ in the closure of $A$ is in the closure of $A$ under pointwise limits of sequences of partial functions. We show that in this case, $B(X)$ is countably Fréchet--Urysohn, that is, each point in the closure of a countable set is a limit of a sequence of elements of that set. This solves a problem of Arnold Miller. The continuous version of this problem is equivalent to a notorious op…

具有不可分测度的可数紧对偶球

Countably tight dual ball with a nonseparable measure

本文在钻石原理假设下构造了一个紧空间K，使得P(K)可数紧但K上存在不可数Maharam型的Radon测度，从而区分了P(K×K)与P(K)×P(K)的紧性并回答了Pol的问题。