We identify the vector space tensor product $\mathbb{R}^{m} \otimes \mathbb{R}^{n}$ with $\mathbb{R}^{mn}$
Let $X$ be the space of all non zero simple tensors $X=\{a\otimes b \mid a\in \mathbb{R}^{n} \setminus \{0\}, \;b\in \mathbb{R}^{m} \setminus \{0\}\}$. Let $PX$ be the projectivization of $X$, that is $PX=\pi (X)$ where $\pi$ is the natural quotient map from $\mathbb{R}^{(mn-1)} \setminus \{0\}$ to $\mathbb{R}P^{mn-1}$.
What can be said about the topology of $X$ and $PX$?Are they prime topological spaces?What can be said about the fixed point property of $X$ and $PX$.
A prime space is a space which is not homeomorphic to a (nontrivial) product spaces.
