Let f be a continuous map of a closed interval into itself, and let
Ω
(
f
)
\Omega (f)
denote the nonwandering set of f. It is shown that if
Ω
(
f
)
\Omega (f)
is finite, then
Ω
(
f
)
\Omega (f)
is the set of periodic points of f. Also, an example is given of a continuous map g, of a compact, connected, metrizable, one-dimensional space, for which
Ω
(
g
)
\Omega (g)
consists of exactly two points, one of which is not periodic.