Prove that $C = \{(x_1,x_2,0)\in R^3 | x_1^2+x_2^2 \leq 1\}$ is closed, dim(C)=2 and find ri(C)

Question

Let $C = \{(x_1,x_2,0)\in R^3 | x_1^2+x_2^2 \leq 1\}$. Show that:

(1) C is closed $\qquad\qquad$ (2) dim(C)=2 $\qquad\qquad$(3) int(C) = $\varnothing$

(3) $ri(C) = \{(x_1,x_2,0)|x_1^2+x_2^2<1\}$

I have no idea on proceed on proof. Can anyone give me some hints? Many thanks

Answer 1

(1) Show that its complement is opened in $\mathbb{R}^3$.

(2) What definition do you want to use? Approach may vary depending on answer. Still, note that dimension does not depend on embedding.

(3) Show that any open ball in $\mathbb{R}^3$ that contains a point of $C$ also contains points not in $C$.

(4) Consider the projection on $\mathbb{R}^2$ which simply forgets the last coordinate.