Is this set perfect?

Question

Let $X$ be a compact and Hausdorff space and $f:X\to X$ is a homeomorphism. Let for all $i\neq j$ and $p\in X$, we have $f^i(p)\neq f^j(p)$. I need to know that whether $\overline{\{f^n(p)\}_{n\in\mathbb{Z}}}$ is uncountable or not. I think that it is sufficient to $\overline{\{f^{n}(p)\}_{n\in\mathbb{Z}}}$ is an infinite and closed without isolated point i.e. it is a perfect set. It is clear that it is infinite and close. Let $p$ be an isolated point, this implies that there is an open set $U$ such that $U=\{p\}$, hence $\{f^n(U)\}_{n\in\mathbb{Z}}$ is an open cover for compact set $\overline{\{f^{n}(p)\}_{n\in\mathbb{Z}}}$. Thus there is $m>0$ with $\overline{\{f^{n}(p)\}_{n\in\mathbb{Z}}}\subseteq \cup_{n=0}^{m}f^n(U)$, that is a contradiction.

Is my proof true? Please help me

Answer 1

There are several problems with your proof:

1. You wrote “Let $p$ be an isolated point”, but $p$ here already has another meaning.
2. Then you write “this implies that there is an open set $U$ such that $U=\{p\}$”. This is wrong. What it implies is that there is an open subset $U$ of $X$ such that $\overline{\{f^n(p)\,|\,n\in\mathbb{Z}\}}\cap U=\{p\}$.