Proof. (1) $\Rightarrow$ (2): See Definition 41.1.14.

(2) $\Rightarrow$ (1): Let $Y = \ol{Gx}$ and $y \in Y$. For any $U \in \cn_{X}(x)$, since $X$ is a compact Hausdorff space, there exists $V \in \cn_{X}(x)$ with $\ol{V}\subset U$. As $N(x, V)$ is syndetic in $G$, there exists $F \subset G$ finite such that $G = F^{-1}N(x, V)$.

Since $\ol{Gx}= Y$, there exists an ultrafilter $\fF \subset 2^{G}$ such that $\fF x \to y$. As $G = \bigcup_{f \in F}N(x, V)$, there exists $f \in F$ such that $f^{-1}N(x, V) \in \fF$. Thus $y = \lim \fF x \subset \ol{f^{-1}N(x, V)}\subset \ol{f^{-1}V}$. Equivalently, $fy \in \ol{V}$, and $Gy \cap U \supset Gy \cao f^{-1}\ol{V}\ne \emptyset$. As this holds for all $U \in \cn_{X}(x)$, $x \in \ol{Gy}$, and $\ol{Gy}= \ol{Gx}= Y$. Therefore every point in $Y$ is transitive, and $G \curvearrowright Y$ is a minimal subflow.$\square$