[Math] Projection, canonical immersion/submersion – are they equivalent, and are they open maps

Question

I am very confused with the concept of projection with the introduction of immersion and submersion. By local immersion/submersion theorem, for a simmersion/submersion $f$, there is is a canonical immersion/submersion locally is equal to $f$.

So does this imply that locally, for a immersion/submersion of $f$ at $x$, it is equal to the projection and inverse projection function?

Are these three all open maps?

Thank you~

Answer 1

Well, projections are open since basic open sets in the product topology are $U \times V$ with $U$ and $V$ open, so projecting down leaves you with $U$ and you're done.

Now, since submersion is locally just a projection, it follows that it is also open.

Immersions need not be open though. For example, take canonical immersion $\mathbb R^1 \to \mathbb R^2$. The domain is surely open but the image is just a line which can't be open in the plane.