Example of an open map on $\mathbb{R}^2$ that is not a submersion

Question

I am searching for an example of a map $f : \mathbb{R}^2 \to \mathbb{R}$ that is open but is not a submersion... I know that any constant map is not a submersion, but it is indeed closed, I am wondering for an example where $f$ is an open map.

I appreciate any help!

@FlybyNight: as far as I know a "submersion" is only defined for manifolds; hence you need the standard topologies on $\mathbb R$ and $\mathbb R^2$. — Cheerful Parsnip, May 03 '16 at 18:56
@FlybyNight, to me, a submersion is a map whose derivative is surjective. — L.F. Cavenaghi, May 03 '16 at 19:41
@GrumpyParsnip So, differentiability only makes sense with respect to the "standard topologies"? — Fly by Night, May 04 '16 at 20:15

score 8 · Accepted Answer · answered May 03 '16 at 18:54

8

Try the map $(x,y) \to x^3.\,\,$

answered May 03 '16 at 18:54

zhw.

105,693

1

this is a good one! I do appreciate. – L.F. Cavenaghi May 03 '16 at 19:42

Example of an open map on $\mathbb{R}^2$ that is not a submersion

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