Let $X$ be a metric space, and let $f:X\rightarrow X$ be a homeomorphism. The $\omega$-limit set of $x\in X$, denoted by $\omega(x,f)$, is the set of cluster points of the forward orbit $\{f^{n}(x)\}_{{n\in\mathbb{N}}}$. Hence, $y\in\omega(x,f)$ if and only if there is a strictly increasing sequence of natural numbers $\{n_{k}\}_{{k\in\mathbb{N}}}$ such that $f^{{n_{k}}}(x)\rightarrow y$ as $k\rightarrow\infty$.

Another way to express this is

The $\alpha$-limit set is defined in a similar fashion, but for the backward orbit; i.e. $\alpha(x,f)=\omega(x,f^{{-1}})$.

Both sets are $f$-invariant, and if $X$ is compact, they are compact and nonempty.

If $\varphi:\mathbb{R}\times X\to X$ is a continuous flow, the definition is similar: $\omega(x,\varphi)$ consists of those elements $y$ of $X$ for which there exists a strictly increasing sequnece $\{t_{n}\}$ of real numbers such that $t_{n}\rightarrow\infty$ and $\varphi(x,t_{n})\rightarrow y$ as $n\rightarrow\infty$. Similarly, $\alpha(x,\varphi)$ is the $\omega$-limit set of the reversed flow (i.e. $\psi(x,t)=\phi(x,-t)$). Again, these sets are invariant and if $X$ is compact they are compact and nonempty. Furthermore,