Atsuji has internally characterized those metric spaces
X
X
for which each real-valued continuous function on
X
X
is uniformly continuous as follows: (1) the set
X
′
X’
of limit points of
X
X
is compact, and (2) for each
ε
>
0
\varepsilon > 0
, the set of points in
X
X
whose distance from
X
′
X’
exceeds
ε
\varepsilon
is uniformly discrete. We obtain these new characterizations: (a) for each metric space
Y
Y
, the Hausdorff metric on
C
(
X
,
Y
)
C\left ( {X,Y} \right )
, induced by a metric on
X
×
Y
X \times Y
compatible with the product uniformity, yields the topology of uniform convergence; (b) there exists a metric space
Y
Y
containing an arc for which the Hausdorff metric on
C
(
X
,
Y
)
C\left ( {X,Y} \right )
yields the topology of uniform convergence; (c) the Hausdorff metric topology on
CL
(
X
)
{\text {CL}}\left ( X \right )
is at least as strong as the Vietoris topology. We also characterize those metric spaces whose hyperspace is such a space.