$ω$ -Rudin空间、良滤确定空间和第一可数空间
$ω$-Rudin spaces, well-filtered determined spaces and first-countable spaces
摘要：
研究了$d$ -空间、良滤空间和Rudin空间的各种可数性质。主要成果包括：（i）如果$T_0$空间$X$的sobrification是一次可数的，则$X$是$ω$ -Rudin空间;（ii）如果每个$ω$ -良滤空间的sobrification是一次可数的，则它是sobrification;（iii）如果一个$T_0$空间是二次可数的或一次可数的，并且有一个可数的底层集，则它是$ω$ -Rudin空间;（iv）第一可数$T_0$空间是良滤确定的;（v）第一可数$ω$ -良滤空间中的不可约闭子集是可数定向的;（vi）第一可数$ω$ -良滤$ω^\ast$ - $d$ -空间是sober的。
英文摘要：
We investigate some versions of $d$-space, well-filtered space and Rudin space concerning various countability properties. The main results include: (i) if the sobrification of a $T_0$ space $X$ is first-countable, then $X$ is an $ω$-Rudin space; (ii) every $ω$-well-filtered space is sober if its sobrification is first-countable; (iii) if a $T_0$ space is second-countable or first-countable and with a countable underlying set, then it is a $ω$-Rudin space; (iv) every first-countable $T_0$ space is well-filtered determined; (v) every irreducible closed subset in a first-countable $ω$-well-filtered space is countably-directed; (vi) every first-countable $ω$-well-filtered $ω^\ast$-$d$-space is 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。
