Show that the union of the unit circle and a spiral is not a submanifold of $\mathbb{R}^2$

Question

Let $M \subset \mathbb{R}$ be defined by

$M = M_{1} \cup M_{2}$, with $M_{1} = \{(x,y) \in \mathbb{R}: x^2 + y^2 =1\}$ and $M_{2}$ is the image of the map $\gamma:(1,\infty) \rightarrow \mathbb{R}, t \mapsto ((1-\frac{1}{t})\cos(t),(1-\frac{1}{t})\sin(t)).$

Show that $M$ is not a submanifold of $\mathbb{R}^2$.

We had to show that the Lemniscate of Gerono is not a submanifold of $\mathbb{R}^2$ in as a previous exercise, which was easy cause it intersected itself. I know the image converges for $t \rightarrow \infty$ to the unit circle. (it looks like a spiral inside the unit circle) But i really don't know how to show that the union of both is not a submanifold. Can someone give me a hint?

Answer 1

Hint:

Take any point on the circle. Can you find an open set about it that intersects the spiral finite number of times. The point being that can you check that any point on the circle has a neighborhood diffeomorphic to $\mathbb{R}$.

Answer 2

Hint: $M$ is similar to the closed topologist's sine curve, which is also not a manifold, because at some points of $M$, it is not even locally connected, let alone being homeomorphic to the Euclidean space.