almost countably subcompact

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 subcompact if there is a π-base B for open sets in X (which is called a subcompact base) such that for every countable subfamily F of B, if F is a regular filter base then F has a non-empty intersection.

Property

• A quasi-regular space X is almost countably subcompact iff there is a π-base P such that every sequence (B_n) in P has a nonempty intersection if cl(B_{n+1}) is contained in B_n.
• Every almost countably subcompact space is Baire.

Remark

• See countably subcompact.

Reference

• Y. Ikeda, Cech-completeness and countable subcompactness, Topology Proc. 14 (1989), 75-87.
• Leszek PiaTkiewicz and Laszlo Zsilinszky, On (strong) α-favorability of the Wijsman hyperspace, Topol. Appl. 157 (2010), 2555-2561.
• Jiling Cao and Heikki J. K. Junnila, Amsterdam Properties of Wijsman hyperspaces, Proc. Amer. Math. Soc. Vol.138, No.2 (2010), pp.769-776.