Simple question about topologies

Question

Let $A=\{(x,y,x+y)|x\in \mathbb{Q},y\in \mathbb{Q}\}$ with the Subspace topology of $\mathbb{R}^3$.

My only question is-does it mean that $A=\mathbb{Q}\times\mathbb{Q}\times\mathbb{Q}$? (Knowing that the sum of two rationals is rational).

Answer 1

No, $A$ is just a proper subspace of $\mathbb{Q}^3$. Note that $(x,x,x)\not\in A$ for any $x\in \mathbb{Q}$ and $x\not=0$.

Answer 2

No. We have $A \subseteq\mathbb{Q}\times\mathbb{Q}\times\mathbb{Q}$ but $A \ne \mathbb{Q}\times\mathbb{Q}\times\mathbb{Q}$.

For example we have $(0,0,1) \in \mathbb{Q}\times\mathbb{Q}\times\mathbb{Q}$, but $(0,0,1) \notin A$.