Let $(X,d)$ be a non-empty compact [metric space](/page/Metric%20Space) with no isolated points, and let $f:X\to X$ be continuous. For $x\in X$, define the forward orbit of $x$ by
1. For every pair of non-empty open sets $U,V\subset X$, there exists $n\in\mathbb{N}_0$ such that $f^n(U)\cap V\ne\varnothing$.
2. There exists $x\in X$ such that $\mathcal{O}_f^+(x)$ is dense in $X$.