Translation-invariant generalized topologies induced by probabilistic norms

Probabilistic Powerdomains and Quasi-Continuous Domains

The probabilistic powerdomain $\mathbf V X$ on a space $X$ is the space of all continuous valuations on $X$. We show that, for every quasi-continuous domain $X$, $\mathbf V X$ is again a quasi-continuous domain, and that the Scott and weak topologies then agree on $\mathbf V X$. This also applies to the subspaces of probability and subprobability valuations on $X$. We also show that the Scott and weak topologies on the $\mathbf V X$ may differ when $X$ is not quasi-continuous, and we give a simple, compact Hausdorff counterexample.

关于向量空间上含向量拓扑的拓扑格的刚性

On Rigidity of Lattice of Topologies Containing Vector Topologies on Vector Spaces

本文证明了向量空间拓扑格中保持向量拓扑的同构必由平移、半线性同构和补映射诱导，从而系数域同构且维数相同，并推广到豪斯多夫情形。

Finite and countable infinite products of Probabilistic Normed Spaces

In this work we first give for PN spaces results parallel to those obtained by Egbert for the product of PM spaces, and generalize results by Alsina and Schweizer in order to study non-trivial products and the product of $m$-- transforms of several PN spaces. In addition we present a detailed study of $α$--simple product PN spaces and, finally, the product topologies in PN spaces which are products of countable families of PN spaces.

Group topologies coarser than the Isbell topology

The Isbell, compact-open and point-open topologies on the set $C(X,\mathbb{R})$ of continuous real-valued maps can be represented as the dual topologies with respect to some collections $α(X)$ of compact families of open subsets of a topological space $X$. Those $α(X)$ for which addition is jointly continuous at the zero function in $C_α(X,\mathbb{R})$ are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections $α(X)$ for which $C_α(X,\mathbb{R})$ is a topological vector space are defined canonically. The Isbell topology coincides with thi…

双绝对连续同胚群上的群拓扑

Group topologies on groups of bi-absolutely continuous homeomorphisms

本文研究双绝对连续同胚群上的群拓扑，证明 τ_ac 与 τ_co 间不存在中间可分群拓扑，并给出 Roelcke 预紧性结果。