  In this article, we consider $\mathcal{I}^\mathcal{K}$-convergence to define
a new concept of convergence namely,
$\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence which generalizes the
notion of $\mathcal{S}$-$\mathcal{I}$-convergence introduced by Guevara et al.
\cite{GSR20} recently. Some properties of
$\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence of sequences and its
relation with compact sets are discussed. In particular, we investigate the
relation between semi-compactness and semi-Lindeloffness by introducing the
notion of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster point of a sequence.
The "Equivalence between semi-dense and dense sets" is utilized to characterize
the set of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster points of a sequence
as semi-closed subsets of a topological space. Moreover, in product space, we
obtain some results for $\mathcal{I}^\mathcal{K}$-convergence which also holds
for $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence.
