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submanifolds

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Let's say you have a curve c going through (0,0) and then it circles back and gets arbitrary close to (0,0) again without actually touching it. The problem isn't with the one "end" of the curve that gets arbitrary close to (0,0), you can always choose a small neighbourhood at every point of the end so that locally, this piece of the curve looks like a line( that is, it's homeomorphic to a line and an immersion)

The problem is with the point (0,0) that is somewhere in the "middle" of your curve: no matter how small you pick your neighbourhood, this neighbourhood will always contain a curve segment from c passing through (0,0) and a second part of c from when it comes arbitrarily close to (0,0) again. Having two parts of the curve, one from the middle and one from the end, in the same neighbourhood is bad and causes the definition to fail here. I don't want to make that argument explicit in a reddit comment, but I hope this gives you a conceptual idea about what to do and where to start.

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