Let $X$ be a Hausdorff space and $f\colon X\to X$ a function. A point $x\in X$ is said to be recurrent (for $f$ if $x\in \omega(x)$ i.e. if $x$ belongs to its $\omega$ limit set. This means that for each neighborhood $U$ of $x$ there exists $n>0$ such that $f^n(x)\in U$

The closure of the set of recurrent points of $f$ is often denoted $R(f)$ and is called the recurrent set of $f$

Every recurrent point is a nonwandering point, hence if $f$ is a homeomorphism and $X$ is compact, $R(f)$ is an invariant subset of $\Omega(f)$ which may be a proper subset.