Remarks on SHD spaces and more divergence properties

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.