On selectively highly divergent spaces

In this paper we introduce the class of SHD spaces, defined by a selection property, and study their fundamental topological features. We exhibit that such class is full of variety: in particular, we construct an SHD space that simultaneously contains a convergent sequence and a dense subspace where the only convergent sequences are the eventually constant ones. We prove that if $X$ is regular and for all $x \in X$ holds $\psi(x, X)>\omega$, then $X_\delta$ is SHD. Finally, for any Hausdorff space $X$ without isolated points, we produce an associated space $s X$ which is extremely disconnected, zero-dimensional, Hausdorff and SHD, and which satisfies $|X|=|s X|, \pi w(X)=\pi w(s X)$ and $c(X)=c(s X)$.