γ-compact

Definition 1

Every γ-open cover has a finite subcover.

Definition 2

Let X be a topological space and let U be a family of open subsets of X. Then X is γ-compact iff every subset family N of X with the properties:

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

Definition 3

Let (X,T) be a topological space. A map γ from T to the power set P(X) of X is called an operation if every G in T is contained in γ(G).
A subset K of X is said to be γ-compact if for every T-open cover C of K there exists a finite subfamily {A_1,...,A_n} of C such that {γ(A_1),...,γ(A_n)} covers K.

Reference

Definition 1

E. Ekici and M. Caldas, Slightly γ-Continuous Functions, Bol. Soc. Paran. Mat. (3s.) Vol.22-2 (2004), pp.63-74.

Definition 2

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

Definition 3

T. Fukutake, On operation-paracompact spaces and products, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 24, 2 (1994), 23-29.