$\omega$-Rudin spaces, well-filtered determined spaces and first-countable spaces

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 $\omega$-Rudin space; (ii) every $\omega$-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 $\omega$-Rudin space; (iv) every first-countable $T_0$ space is well-filtered determined; (v) every irreducible closed subset in a first-countable $\omega$-well-filtered space is countably-directed; (vi) every first-countable $\omega$-well-filtered $\omega^\ast$-$d$-space is sober.