almost countably base-compact

Definition

A quasi-regular space X (i.e. for every nonempty open subset U, there is a nonempty open subset V whose closure is contained in U) is called almost countably base-compact if there is a π-base B of X such that for every countable subfamily C of B with fip, the closures of the members of C have a nonempty intersection.

Remark

• Every almost countably base-compact space is almost countably subcompact

Reference

• Jiling Cao and Heikki J. K. Junnila, Amsterdam Properties of Wijsman hyperspaces, Proc. Amer. Math. Soc. Vol.138, No.2 (2010), pp.769-776.