  Given a metric continuum $X$, a nonempty proper closed subspace $B$ of $X$,
does not block a point $p\in X\setminus B$ provided that the union of all
subcontinua of $X$ containing $p$ and contained in $X\setminus B$ is a dense
subset of $X$. The collection of all nonempty proper closed subspaces $B$ of
$X$ such that $B$ does not block any element of $X\setminus B$ is denoted by
$NB(F_{1}(X))$. In this paper we prove that for each completely metrizable and
separable space $Z$, there exists a continuum $X$ such that $Z$ is homeomorphic
to $NB(F_{1}(X))$. This answers a series of questions by Camargo, Capul\'in,
Casta\=neda-Alvarado and Maya.
