  In the paper we study algebraic property of the monoid
$\mathbf{I}\mathbb{N}_{\infty}^{\textbf{g}[j]}$ of cofinite partial isometries
of the set of positive integers $\mathbb{N}$ with the bounded finite noise $j$.
We extend the Eberhart and Selden results for the closure of the bicyclic
monoid onto $\mathbf{I}\mathbb{N}_{\infty}^{\textbf{g}[j]}$ for any positive
integer $j$. In particular we show that for any positive integer $j$ every
Hausdorff shift-continuous topology $\tau$ on
$\mathbf{I}\mathbb{N}_{\infty}^{\textbf{g}[j]}$ is discrete and if
$\mathbf{I}\mathbb{N}_{\infty}^{\textbf{\emph{g}}[j]}$ is a proper dense
subsemigroup of a Hausdorff semitopological semigroup $S$, then $S\setminus
\mathbf{I}\mathbb{N}_{\infty}^{\textbf{\emph{g}}[j]}$ is a closed ideal of $S$,
and moreover if $S$ is a topological inverse semigroup then $S\setminus
\mathbf{I}\mathbb{N}_{\infty}^{\textbf{\emph{g}}[j]}$ is a topological group.
Also we describe the algebraic and topological structure of the closure of the
monoid $\mathbf{I}\mathbb{N}_{\infty}^{\textbf{\emph{g}}[j]}$ in a locally
compact topological inverse semigroup.
