The "figure eight" as the image of the injective immersion $g: \mathbb{R} \rightarrow \mathbb{R}^2$, dada por $g(t) = (2 \cos(\pi/2 + 2 \arctan(t)), \sin 2 (\pi/2 + 2 \arctan(t)))$. It's different from the subset $A$ of points in $\mathbb{R}^2$ wich drawn the "figure eight". As an immersed submanifold, "figure eight" has its topology induced from $\mathbb{R}$ while $A$, as a subset of $\mathbb{R}^2$ has the topology induced from this last space.
We know that $A$ is a closed subset of $\mathbb{R}^2$, but we can't think about $g(R)$ as we think about $A$, because they have different structures. For example, $g: \mathbb{R} \rightarrow g(R)$ is an homeomorphism with the topology induced from $\mathbb{R}$, but it's not even a submanifold with the topology induced from $\mathbb{R}^2$.
Every immersed submanifold is a completely strange object in the ambient space, it has its own topology, inherited from its origin space, and it has nothing to do with the ambient space.
Once everything I wrote above is right (if there is any misunderstood, please correct me) my doubts are is:
(1) How can I check if an immersed submanifold is closed in the ambient space? This makes no sense to me!
(2) How can I show the "figure eight" is not a closed submanifold?
(3) If you could, at least, give me a proof tha some immersed submanifold is closed (or not closed), it will help me a lot! I don't know how can I deal with theses topologies.
Thank you very much and please forgive any english mistakes (I'm brasilian).
