please help me to find answer this question,
Why can't we write, $\mathbb{R}P^{n}=\mathbb{R}P\times \cdots \times \mathbb{R}P$
if $\mathbb{R}P^{n}$ is real projective space?
I know (in geometry) $\mathbb{R}P^{n}$ is projective space of $\mathbb{R}^{n+1}$. it is real line if space is $\mathbb{R}^{2}$ and is a real plane if space is $\mathbb{R}^{3}$ and go on.
I also know (in algebraic geometry) this space is equivalence classes of this relation
$(x_1,\ldots , x_n) \sim (y_1,\ldots , y_n) \longleftrightarrow (x_1,\ldots , x_n)=\lambda (y_1,\ldots , y_n) $
such that $\lambda \neq 0$
As topological spaces, for example, $\Bbb RP^2\neq\Bbb RP^1\times\Bbb RP^1$. Note that $\Bbb RP^1$ is just the circle $S^1$, so $\Bbb RP^1\times\Bbb RP^1$ is a torus. On the other hand $\Bbb RP^2$ is a nonorientable surface.
Edit: a more elementary answer is that $\Bbb RP^2\neq\Bbb RP^1\times\Bbb RP^1$ as sets. We often write, for example, $\Bbb R^3=\Bbb R\times\Bbb R^2$: even though elements of $\Bbb R\times\Bbb R^2$ are technically pairs $(a,(b,c))$, there is an obvious enough way to ignore parenthesis and identify this element with $(a,b,c)\in\Bbb R^3$. On the other hand, you cannot take an element $[a:b:c]\in\Bbb RP^3$ and consider it as an element $([a],[b:c])\in\Bbb RP^1\times\Bbb RP^2$. For example, consider $[0:0:1]$.