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The definition of smooth immersed submanifold is from Lee's Trilogy. The topology of the smooth immersed submanifold $S$ of $M$ is such that the inclusion $\iota: S\to M$ is a smooth immersion. However, I don't know how to use this definition to say what exactly their open sets are. Could you give me any advice? The following is the question which I'm trying to solve using this concept.

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    By definition, immersed submanifold $S\subseteq M$ is an immersion $\iota:S_0\mapsto M$, and it is the topology of $S_0$ that is induced on $S$(As you mentioned). And there is really nothing to do with the specific topology on $S_0$; As $\iota$ is smooth, it is continuous, thus any inverse image of an open set in $M$ is open. – cjackal Nov 10 '15 at 05:43
  • Thanks for your answer. Your advice enabled me to solve the exercise! – Math.StackExchange Nov 10 '15 at 06:02

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