$X$ is a metric space and $f: X \rightarrow X$ is a dynamical system.
Prove: $w(x_{0})$ is closed.
Here the set $w(x_{0})$ is the future of the orbit of $x_0$, defined as $$\omega(x_0) = \{y \mid \exists (n_k)\, n_k \to \infty, f^{n_k}(x_0) \to y\}$$