Metrizability and certain generalized metrizability on hyperspaces with the locally finite topology

For a topological space X, let $CL\left(X\right)$ be the set of all nonempty closed of X, and denote the set $CL\left(X\right)$ with the locally finite topology by $\left(CL\right(X),\text{LF})$.

In the first part of this paper, generalized metric spaces properties such as semi-stratifiability and Lašnev space property etc on hyperspaces with the locally finite topology are investigated and some metrization theorems are established.

(1) If X is a normal $D_0$-space, then $\left(CL\right(X),\text{LF})$ is semi-stratifiable if and only if $\left(CL\right(X),\text{LF})$ is metrizable.

(2) If X is a metrizable space, then $\left(CL\right(X),\text{LF})$ is of countable tightness if and only if $\left(CL\right(X),\text{LF})$ is metrizable.

(3) $\left(CL\right(X),\text{LF})$ is a Lašnev space if and only if it is metrizable.

(4) $(MA+\neg CH)$ $\left(CL\right(X),\text{LF})$ is a $D_0$-space if and only if it is metrizable.

In the second part, the first step is made towards the investigation of hyperspaces with the locally finite topology with an ${\omega }^{\omega }$-base. The following results are obtained.

(5) If $\left(CL\right(X),\text{LF})$ is a Fréchet-Urysohn space with an ${\omega }^{\omega }$-base, then $\left(CL\right(X),\text{LF})$ is first-countable.

(6) Let X be metrizable. If $\left(CL\right(X),\text{LF})$ has an ${\omega }^{\omega }$-base, then the subspace consisting of all non-isolated points of the space X is separable and X has an ${\omega }^{\omega }$-base at each closed subset of X.