Let K be a compact metric space. A homeomorphism
f
:
K
∣
f:\,K\mid
is expansive if there exists
ε
>
0
\varepsilon \, > \,0
such that if
x
,
y
∈
K
x, y\, \in \,K
satisfy
d
(
f
n
(
x
)
,
f
n
(
y
)
)
>
ε
d\left ( {{f^n}\left ( x \right ),\,{f^n}\left ( y \right )} \right )\, > \,\varepsilon
for all
n
∈
Z
n\, \in \,{\textbf {Z}}
(where
d
(
⋅
,
⋅
)
d\left ( { \cdot ,\, \cdot } \right )
denotes the metric on K) then
x
=
y
x\, = \,y
. We prove that a compact metric space that admits an expansive homeomorphism is finite dimensional and that every minimal set of an expansive homeomorphism is 0-dimensional.