I'm looking for some streamlined ways to verify that certain maps are topological embeddings, since these pop up all the time in examples. For example, I was able to prove that the helix map $\mathbb{R} \to \mathbb{R}^3$ given by $t \mapsto (\cos2\pi t, \sin 2\pi t, t)$ is an embedding by checking that it's closed (I proved this by showing that the complements of open intervals are taken to closed sets in $\mathbb{R}^3$) but this map is so obviously an embedding in my intuition that I feel like there must be a cleaner way to prove it. I'm looking for something similar to the result that any injective continuous map of a compact space into a Hausdorff space is automatically an embedding.
In general, what are some easy techniques to prove that a map is an embedding in practice? I always find myself trying to prove that the map is open or closed, which isn't always that straightforward.
Best Answer
The given function is continuous.
As it is a bijection from R onto a helix it has an inverse.
Show the inverse is continuous.
From that, it's apparent it is an embedding.