Proof that $N$ is an immersed submanifold.

Question

Consider the following set in $\mathbb{R}^3$: $$N := \{(\cos(t),\sin(t),e^t) \rvert t \in \mathbb{R}\} \sqcup \{(\cos(t),\sin(t),0) \vert t \in \mathbb{R}\}.$$ This is the union of a 'spiral' and a circle. We have to show (or disprove) that this is an immersed submanifold. Intuitively, I would guess this is indeed an immersed submanifold. We equip this with the topology whose opens are 'along' the spiral and the circle. Then, the inclusion is evidently smooth and furthermore the differential is injective. However, I'm not entirely sure if this is truly correct.

Answer 1

An immersed submanifold of a manifold $M$ is the image $N = f(S)$ of an immersion $f: S \to M$ defined on a smooth manifold $S$. Note that in general $N$ is not a submanifold of $M$, even if $f$ is injective (which is in general not true).

So let $S = \mathbb R \times \{0,1\} \subset \mathbb R^2$. This is a smooth manifold (in fact, a smooth submanifld of $\mathbb R^2$). Define $f : S \to \mathbb R^3, f(t,0) = (\cos t, \sin t, e^t), f(t,1) = (\cos t, \sin t, 0)$. This is clearly an immersion with image $N$ as in your question. However, it is not a submanifold of $\mathbb R^3$.

Alternatively you can define $S = \mathbb R \times \{(0,1)\} \cup S^1 \times \{0\} \subset \mathbb R^3$, where $S^1 = \{ (x,y) \mid x^2 + y^2 = 1 \}$, and $f(t,0,1) = (\cos t, \sin t, e^t), f(p,0) = (p,0)$. Then $f$ is an injective immersion and its restrictions to $\mathbb R \times \{(0,1)\}$ and to $S^1 \times \{0\}$ are embeddings.