*Definition 1 [#x8a14ed2]
Every [[γ-open]] cover has a finite subcover.

*Definition 2 [#ibc35a48]
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:
+for A in U, we can pick some R in N such that A or the complement of A is contained in R
+the interiors of members of N cover X
has a finite subfamily which covers X.

*Definition 3 [#gb61c59d]
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 [#tf41ac9f]
: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.