Some equivalences for Martin's Axiom in asymmetric topology

Separation axioms and covering dimension of asymmetric normed spaces

In this paper, we approach the question if some of the separation axioms are equivalent in the class of asymmetric normed spaces. In particular, we make a remark on a known theorem which states that every $T_1$ asymmetric normed space with compact closed unit ball must be finite-dimensional. We also explore the product structure of these spaces and characterize the topological (covering) dimension of all finite-dimensional asymmetric normed spaces.

A countably compact topological group with the non-countably pracompact square

Under Martin's Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.

非OCA与Fréchet空间的乘积

Not OCA and products of Frechet spaces

本文继续研究两个可数Fréchet空间乘积的M-可分性，重点考察马丁公理下的情形，并指出OCA蕴含肯定答案，而已有独立性结果仅适用于$\mathfrak c\leqω_2$的模型。

The topology of ultrafilters as subspaces of $2^ω$

Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martin's Axiom for countable posets, to distinguish non-principal ultrafilters on $ω$ up to homeomorphism. Here, we identify ultrafilters with subpaces of $2^ω$ in the obvious way. Using the same methods, still under Martin's Axiom for countable posets, we will construct a non-principal ultrafilter $\UU\subseteq 2^ω$ such that $\UU^ω$ is countable dense homogeneous. This consistently answers a question of Hrušák and Zamora Avilés. Finally, we will give some partial resu…

Products and countable dense homogeneity

Building on work of Baldwin and Beaudoin, assuming Martin's Axiom, we construct a zero-dimensional separable metrizable space $X$ such that $X$ is countable dense homogeneous while $X^2$ is not. It follows from results of Hrušák and Zamora Avilés that such a space $X$ cannot be Borel. Furthermore, $X$ can be made homogeneous and completely Baire as well.

Cosmic dimensions

Martin's Axiom for $σ$-centered partial orders implies that there is a cosmic space with non-coinciding dimensions.