My question is related to this question, i.e. under which conditions we have that $\mu$-a.e. point has a dense orbit in a dynamical system $(X,T)$ with measure $\mu$. I am confused about the assumptions. In at least two textbooks (An Introduction to Ergodic Theory, p.29; Introduction to Dynamical Systems, p.76) I read that T must be continuous, but I don't see where the continuity of $T$ comes in.
The statement: Let $X$ be a compact metric space, $T:X\rightarrow X$ a continuous map and $\mu$ a $T$-invariant borel measure on X. If $T$ is ergodic, then the orbit of $\mu$-a.e. point is dense in $X$.
This can be proven by showing that the set of all points whose forwars orbit visit every element of an open basis has full measure. Let $\{U_i\}_{i\in\mathbb{N}}$ be a countable open basis for X, then for every $U_i$, $\mu(U_i)>0$ and $T^{-1}(\bigcup_{k\in \mathbb{N}}T^{-k}U_i)\subseteq \bigcup_{k\in \mathbb{N}}T^{-k}U_i$. Since $T$ is measure-preserving and ergodic, it follows that $\mu(\bigcup_{k\in \mathbb{N}}T^{-k}U_i)$ has full measure in X.
Where exactly is the continuity of $T$ used? I would appreciate your help!