Open, closed, neither or both in $\mathbb R^2$?

Question

$$\{ (x,y)\in\mathbb R^2: \exp(x^2+y^2) = 1+ (y^3-x^3)(x^7+y^7) \}$$

I usually tell if something is open or closed thinking geometrically. Would I be expected to think about what this looks like? Or is there another way to tell?

Thank you.

score 5 · Answer 1 · answered Sep 01 '14 at 17:27

5

Hint: inverse image of a closed set under a continuous function.

answered Sep 01 '14 at 17:27

Robert Israel

448,999

score 1 · Answer 2 · answered Sep 01 '14 at 17:27

1

Consider the function $f(x,y) = \exp(x^2+y^2)-(y^3-x^3)(x^7+y^7)-1$. The set you are concerned with is the preimage of $0$ under a continuous function.

answered Sep 01 '14 at 17:27

Gyu Eun Lee

18,793

Open, closed, neither or both in $\mathbb R^2$?

2 Answers2