If
A
A
and
B
B
are nonvoid subsets of a metric space
(
X
,
d
)
(X,d)
, the set of points
x
∈
X
x \in X
for which
d
(
x
,
A
)
=
d
(
x
,
B
)
d(x,A) = d(x,B)
is called the equidistant set determined by
A
A
and
B
B
. Among other results, it is shown that if
A
A
and
B
B
are connected and
X
X
is Euclidean
n
n
-space, then the equidistant set determined by
A
A
and
B
B
is connected.