  It is well known that for a Hausdorff topological group $X$, the limits of
convergent sequences in $X$ define a function denoted by $\lim$ from the set of
all convergent sequences in $X$ to $X$. This notion has been modified by Connor
and Grosse-Erdmann for real functions by replacing $\lim$ with an arbitrary
linear functional $G$ defined on a linear subspace of the vector space of all
real sequences. Recently some authors have extended the concept to the
topological group setting and introduced the concepts of $G$-continuity,
$G$-compactness and $G$-connectedness. In this paper we prove some results on
different types of $G$-compactness for topological group with operations which
include topological groups, topological rings without identity, R-modules, Lie
algebras, Jordan algebras, and many others.
