First-countability, ω-Rudin spaces and well-filtered determined spaces

Abstract In this paper, we investigate some versions of d-space, well-filtered space and Rudin space concerning various countability properties. It is proved that every T 0 space with a first-countable sobrification is an ω-Rudin space and every first-countable T 0 space is well-filtered determined. Therefore, every ω-well-filtered space with a first-countable sobrification is sober. It is also shown that every irreducible closed subset in a first-countable ω-well-filtered space is countably directed, hence every first-countable ω ⁎ -well-filtered d-space is sober.