Let $X$ be a compact metric space and let $T:X \to X$ be a homeomorphism.
a) When do you say that $T$ is minimal?
b) When do you say that a subset $A$ of $X$ is minimal?
c)Prove that $T$ has a minimal subset.
d)Give an example of $X$ and $T$ such that $T$ is not minimal and which has at least countably many minimal sets. List countable many minimal sets of $T$.
Here a) $T$ is said to be minimal if $\overline{O_T(x)}=X; \forall x \in X$ where $O_T(x)$ is the orbit of $x$ in $X$.
b) A set $A$ is s.t.b minimal if it is closed under the action of $T$ and $T|_A: A \to A$ is minimal.
For c) I need proof verification. c)I think $O_T(x)$ is itself a minimal set here. Because $T$ is a homeomorphism. So, $O_T(x)$ is closed and $T|_{O_T(x)}$ is minimal.
d) I ask for an example here. I don't know the answer.
Your part c) is absolutely correct.
For part d) consider $X=\{\frac 1 n| n\in \Bbb N \}\cup \{0\}$. Then consider $$T:X \to X$$ such that $T(0)=0$ and $T(\frac 1 n)=\frac 1 {n+1}$. Then $T$ is not clearly minimal as $O_T(0)=\{0\}$ but it has countable minimal subsets. You check why?