Is $\mathbb{R}P^1 \times \mathbb{R}P^1 = \mathbb{R}P^2$? If it isn't what does it look like?

Question

We know that $\mathbb{R}P^1$ may be conceptualized as the set of all lines through the origin in $\mathbb{R}^2$. Alternatively, it may be conceptualized as a hemisphere of $S^1$ with antipodal points at the equator identified. How to think of $\mathbb{R}P^1 \times \mathbb{R}P^1$?

Answer 1

No. $\Bbb RP^1$ is homeomorphic to a circle, which makes $\Bbb RP^1\times \Bbb RP^1$ into a torus, which is definitely not homeomorphic to $\Bbb RP^2$