H-set

Definition 1

• A subset A of a space X is an H-set in X if for each open cover C of A, there is a finite subfamily F of C such that the closures of members of F cover A.

Definition 2?

• A nonempty subset S of a space X is an H-set (of X ) if every open filter on X which meets S, has a cluster point in S.

Reference

Definiton 1

R.F. Dickman, Jr. and J.R. Porter, Between minimal Hausdorff and compact Hausdorff spaces, Topology Proc. Vol.9 (1984), p.243-268.

Definition 2

Krystock, Robert L., Adherent compact spaces. (English), Proc. Am. Math. Soc. 107, No.4, 1117-1125 (1989).