A metric space
(
X
,
d
)
\left ( {X,d} \right )
is called an
M
M
-space if for every
x
x
and
y
y
in
X
X
and for every
r
∈
[
0
,
λ
]
r \in \left [ {0,\lambda } \right ]
we have
B
[
x
,
r
]
∩
B
[
y
,
λ
−
r
]
=
{
z
}
B\left [ {x,r} \right ] \cap B\left [ {y,\lambda - r} \right ] = \left \{ z \right \}
for some
z
∈
X
z \in X
, where
λ
=
d
(
x
,
y
)
\lambda = d\left ( {x,y} \right )
. It is the object of this paper to study
M
M
-spaces in terms of proximinality properties of certain sets.