First countability, ω-well-filtered spaces and reflections

Some open problems on well-filtered spaces and sober spaces

“…This problem was first solved by Lawson and Xi [35], and later answered by Xu, Shen, Xi and Zhao [51,52] using a different method.…”

D-completion, well-filterification and sobrification

“…Definition 2.6. [14] A T 0 space X is called ω-well-filtered, if for any countable filtered family [14] Let X be a T 0 space. A nonempty subset A ∈ KF ω (X) if and only if there exists a countable filtered family K ⊆ Q(X) such that cl(A) is a minimal closed set that intersects all members of K. The set of all closed KF ω -subsets of X is denoted by KF ω (X).…”

“…Definition 2.8. [14] A subset A of a T 0 space X is called a ω-well-filtered determined set, W D ω set for short, if for any continuous mapping f : X −→ Y to a ω-well-filtered space Y , there exists a unique y A ∈ Y such that cl(f (A)) = cl({y A }). The set of all closed ω-well-filtered determined subsets of X is denoted by WD ω (X).…”

“…Proof. It suffices to prove that the well-filterification of X is second countable by Theorem 4.1 in [14]. Let B be a countable basis of X, U = {♦U | U ∈ B}.…”

“…Lawson, Wu and Xi gave the sufficient condition that the space X is core-compact for a well-filtered space X to be sober in [8]. Xu and Shen proved that every first countable well-filtered space is sober in [14]. Xi and Lawson obtained the result that every monotone convergence space X with the property that ↓(K ∩ A) is a closed subset of X for any closed subset A and compact saturated set K is well-filtered in [12].…”