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Let $G$ be a compact group acting continuously and freely on a compact Hausdorff space $X$. Assume that $(g_\lambda)_{\lambda\in\Lambda}$ and $(x_\lambda)_{\lambda\in\Lambda}$ are nets in $G$ and $X$ respectively s.t. $(x_\lambda)_{\lambda\in\Lambda}$ converges to some $x\in X$ and $(g_\lambda\cdot x_\lambda)_{\lambda\in\Lambda}$ converges (in $X$) to $x$. Does it follow that $(g_\lambda)_{\lambda\in\Lambda}$ converges to $e$, the unit of the group $G$?

Another thing is that from compactness of $G$, there exists a convergent subnet of $(g_\lambda)_{\lambda\in\Lambda}$, and then using freeness of the action, one can check easily that the subnet converges to $e$, but of course- it is not enough.

Thanks

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    $\begingroup$ Hint: I suspect that your group $G$ is also Hausdorff. In this case, argue by contradiction. Suppose that $(g_\lambda)$ does not converge to $e$. Then, by compactness, there is a subnet in it which converges to some $g\ne e$. Then argue that $g$ fixes $x$. $\endgroup$ – Moishe Kohan Feb 16 '18 at 14:06

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