关于强d-空间和超H-sober空间的几个公开问题
On some open problems concerning strong d-spaces and super H-sober spaces
摘要：
在本文中，我们强调了徐和赵所提出的一些开放性问题。特别地，我们专注于强$d$ -空间和回答两个开放的问题，有关强$d$ -空间。一个是关于任意强$d$ -空间族的乘积空间。另一个是关于强$d$ -空间范畴$\mathbf{S}$ - $\mathbf{Top}_{d}$的反射性。同时，对于不可约子集系统（简称$R$ -子集系统）$\mathrm{H}$和$T_{0}$空间$X$，证明了$\mathrm{H}$自然满足性质$M$ .在此基础上，我们推出了$X$是超$\mathrm{H}$ -清醒的当且仅当$X$是$\mathrm{H}$ -清醒的且$\mathrm{H}$满足性质$Q$ .并对徐在\cite{Xiaoquan21}中提出的一些问题作了肯定的回答。进一步得到了超清醒空间的有限积是超清醒的。
英文摘要：
In this paper, we highlight some open problems stated by Xu and Zhao. In particular, we focus on strong $d$-spaces and answer two open problems concerning strong $d$-spaces. One is about the product space of an arbitrary family of strong $d$-spaces. The other concerns the reflectivity of the category $\mathbf{S}$-$\mathbf{Top}_{d}$ of strong $d$-spaces. Meantime, for an irreducible subset system ($R$-subset system for short) $\mathrm{H}$ and a $T_{0}$ space $X$, we prove that $\mathrm{H}$ naturally satisfies property $M$. Based on this, we deduce that $X$ is super $\mathrm{H}$-sober iff $X$ is $\mathrm{H}$-sober and $\mathrm{H}$ satisfies property $Q$. Additionally, we give positive answers to some questions posed by Xu in \cite{Xiaoquan21}. Furthermore, we obtain that the finite product of hyper-sober spaces is hyper-sober.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
