  For a $T_0$ space $X$, let $\mk (X)$ be the poset of all compact saturated
sets of $X$ with the reverse inclusion order. The space $X$ is said to have
property Q if for any $K_1, K_2\in \mk (X)$, $K_2\ll K_1$ in $\mk (X)$ if{}f
$K_2\subseteq \ii~\!K_1$. In this paper, we give several connections among the
well-filteredness of $X$, the sobriety of $X$, the local compactness of $X$,
the core compactness of $X$, the property Q of $X$, the coincidence of the
upper Vietoris topology and Scott topology on $\mk (X)$, and the continuity of
$x\mapsto\ua x : X \longrightarrow \Sigma~\!\! \mk (X)$ (where $\Sigma~\!\! \mk
(X)$ is the Scott space of $\mk (X)$). It is shown that for a well-filtered
space $X$ for which its Smyth power space $P_S(X)$ is first-countable, the
following three properties are equivalent: the local compactness of $X$, the
core compactness of $X$ and the continuity of $\mk (X)$. It is also proved that
for a first-countable $T_0$ space $X$ in which the set of minimal elements of
$K$ is countable for any compact saturated subset $K$ of $X$, the Smyth power
space $P_S(X)$ is first-countable. For the Alexandroff double circle $Y$, which
is Hausdorff and first-countable, we show that its Smyth power space $P_S(Y)$
is not first-countable.
