α-compact

Definition

A topological space is called α-compact iff every α-open cover has a finite subcover.

Definition 2

A topological space X is called α-compact iff every subset family N with the properties:

1. the interiors of members of N cover X
2. if Y is open we can pick A in N such that Y or the complement of Y is contained in A

has a finite subfamily which covers X.

Reference

Definition 1

D.S. Jankovic, I.L. Reilly and M.K. Vamanamurthy, On strongly compact topological spaces, Questions Answers Gen. Topology 6 (1988), no.1, 29-40.
J. Dontchev, M. Ganster and T. Noiri, On p-closed spaces, Internat. J. Math. & Math. Sci. Vol.24, No.3 (2000) pp.203-212.

Definition 2

D.V.Thampuran, Nets and Compactness, Portugaliae Mathematica Vol.28(1) pp.37-54.