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## A Closed Immersed Submanifold [closed]

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).

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For question (1): Whether a set $X$ is closed in an ambient space is a property of the set $X$; giving it a different topology won't affect whether it's closed. For question (2) The trouble is not with "closed" but with "submanifold". – Andreas Blass Sep 29 at 20:27
Image of an embedding of a manifold need not be a manifold, the standard requirement to add is that the embedding is proper. You should review some point-set topology before trying differential topology. In particular, read about the difference between continuous bijection and homeomorphism. – Misha Sep 29 at 21:39
For (2), you just need to observe that the tangent directions to $A$ at $(0,0)$ do not all lie on a line as they would on a submanifold. Instead, they lie on the union of two transverse lines, an "X". – Kevin Kordek Sep 29 at 23:11
I think that math.stackexchange.com would be more suitable for your question. Please read mathoverflow.net/faq , and you will learn that MO is for research level math questions. – BS Sep 30 at 10:06
Keep in mind that some terms in this area are overloaded with different meanings. For example, "closed manifold" usually means a compact manifold without boundary and this has nothing to do with closed sets in a topological space. You should make it clear for yourself what meaning of "closed" you mean. – Sergei Ivanov Sep 30 at 11:42