δ-compact

Definition

Every δ-open cover has a finite subcover.

Definition 2

Let X be a topological space and let B be a cover of X. Then X is δ-compact iff every subset family N of X with the properties:

1. for any A in B, we can pick some R in N such that A or the complement of A is contained in R
2. if U is the topology generated by B, the U-interiors of members of N cover X
   has a finite subfamily which covers X.

Reference

Definition 1

Raja Mohammad Latif, Topological Properties of δ-Open Sets, King Fahd University of Petroleum & Minerals, Department of Mathematical Sciences, Technical Report Series, TR409 (2009)

Definition 2

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