Let $(X,d)$ be a non-empty compact [metric space](/page/Metric%20Space), let $f: X \to X$ be continuous, and for $n \in \mathbb{N}_0$ let $f^n: X \to X$ denote the $n$-fold iterate of $f$, with $f^0=\operatorname{id}_X$. For $x \in X$, define the omega-limit set of $x$ by
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\begin{align*}
\omega_f(x) := \{y \in X : \text{there exists a strictly increasing sequence } (n_k)_{k=1}^{\infty} \subset \mathbb{N}_0 \text{ such that } f^{n_k}(x) \to y\}.
\end{align*}
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Then $\omega_f(x)$ is non-empty, compact, and forward invariant, meaning $f(\omega_f(x)) \subset \omega_f(x)$. If $f$ is a homeomorphism, then $\omega_f(x)$ is invariant, meaning $f(\omega_f(x))=\omega_f(x)$.