Please can you help me to understand how the dimension of the set of separable states is $\dim \cal H_1 + \dim \cal H_2$?
This is the relevant passage:
So far, we have assumed implicitly that the system is made of a single component. Suppose a system is made of two components; one lives in a Hilbert space $\cal H_1$ and the other in another Hilbert space $\cal H_2$. A system composed of two separate components is called bipartite. Then the system as a whole lives in a Hilbert space $\cal H = \cal H_1 \otimes \cal H_2$, whose general vector is written as $$\left|\, \psi \right\rangle = \sum_{i,j} c_{ij} \left|\,e_{1,i}\right \rangle \otimes \left|\,e_{2,j}\right\rangle, \tag{2.29}$$ where $\{|\,e_{a,i}\rangle\}$ ($a=1,2$) is an orthonormal basis in $\cal H_a$ and $\sum_{i,j} |c_{ij}|^2 = 1$.
A state $|\,\psi \rangle \in \cal H$ written as a tensor product of two vectors as $|\,\psi \rangle = |\,\psi_1 \rangle \otimes |\,\psi_2\rangle$, ($|\,\psi_a\rangle \in \cal H_a$) is called a separable state or a tensor product state. A separable state admits a classical interpretation such as “The first system is in the state $|\,\psi_1\rangle$, while the second system is in $|\,\psi_2\rangle$.” It is clear that the set of separable states has dimension $\dim \cal H_1 + \dim \cal H_2$.
