I just have been introduced to the axiom "defining" bosons and fermions, namely (for fermions): Consider a collection of $N$ identical particles moving in $\mathbb{R}^3$ and having a (half-integer) spin $l$. For each $\psi \in L^2(\mathbb{R}^{3N};(V_l)^{\otimes N})$ and each permutation $\sigma$, we have $\psi(x^{\sigma(1)},\ldots, x^{\sigma(N)})=sgn(\sigma)\psi(x^1,\ldots, x^N)$.
Then, as an example of my book, we have a look at Pauli's exclusion principle for two electrons. It is stated that the function $\Psi: \mathbb{R}^6 \rightarrow \mathbb{C}^2 \otimes \mathbb{C}^2$ given by $\Psi(x^1,x^2)=\psi(x^1)\otimes \psi(x^2)$ is not a possible state of a two-electron system, since $\Psi$ does not satisfy the above property.
Hence, we must have $\Psi(x^2,x^1)=\psi(x^2)\otimes \psi(x^1)\stackrel{!}{=}\psi(x^1)\otimes \psi(x^2)=\Psi(x^1,x^2) \neq sgn(P_{12})\Psi(x^1,x^2)$.
It has to be obvious but I don't get why the middle equality holds. This is probably because, as a non-physicist, the only intuition I have behind this tensor product is something like "we define a state of a many-particle system by considering the states each particle is in". I would appreciate a hand with that.
