Is the following $f:S^2 \rightarrow \mathbb{R}^4$ an embedding?

Question

I have $f:S^2 \rightarrow \mathbb{R}^4$ given by $$f(x_1,x_2,x_3) = \frac{(x_1, x_1x_3, x_2, x_2x_3)}{1+x_3^2}$$ If I am not wrong, the map above is an injective immersion. I wanted to check if it is or not an embedding.

I argued that f is not an embedding since it is not a homeomorphism onto its image. Is it the correct way of reasoning?

Answer 1

There is a classical result saying this:

A proper injective immersion is an embedding.

As $S^2$ is compact and $f$ continuous, if $B$ is compact in $\mathbb{R}^4$, $f^{-1}(B)$ is closed in $S^2$, hence compact. This says that $f$ is proper.

Thus, one cannot expect $f$ to be an injective immersion without it being an embedding: your reasoning isn't correct.