# Induced action of topological groups

Let $G$ be a polish group, $H$ be an open subgroup of $G$ and $X$ be any metric space on which $G$ act. I want to show the following fact:

If the restriction to $H$of the action of $G$ on $X$ is continuous, then the action of $G$ on $X$ is continuous.

Thank for any help

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This can be shown using the same trick as the one I used in the answer to your other question, i.e., testing sequential continuity, exploiting that $G/H$ is discrete. –  Martin Dec 29 '12 at 13:53