  In this paper, we analyze the existence of algebraic and topological
structures in the set of sequences that contain only a finite number of zero
coordinates. Inspired by the work of Daniel Cariello and Juan B.
Seoane-Sep\'ulveda, our research reveals new insights and complements their
notable results beyond the classical \( \ell_p \) spaces for \( p \) in the
interval from 1 to infinity, including the intriguing case where \( p \) is
between 0 and 1.
  Our exploration employs notions such as S-lineability, pointwise lineability,
and (alpha, beta)-spaceability. This investigation allowed us to verify, for
instance, that the set \( F \setminus Z(F) \), where \( F \) is a closed
subspace of \( \ell_p \) containing \( c_0 \), is (alpha, c)-spaceable if and
only if alpha is finite.
