  The class of SHD spaces was recently introduced in [12]. The first part of
this paper focuses on answering most of the questions presented in that
article. For instance, we exhibit an example of a non-SHD Tychonoff space $X$
such that $\mathscr{F}[X]$, the Pixley-Roy hyperspace of $X$, $\beta X$, the
Stone-\v{C}ech compactification of $X$, and $C_p(X)$, the ring of continuous
functions over $X$ equipped with the topology of pointwise convergence, are
SHD.
  In the second part of this work we will present some variations of the SHD
notion, namely, the WSHD property and the OHD property. Furthermore, we will
pay special attention to the relationships between $X$ and $\mathscr{F}[X]$
regarding these new concepts.
