A few questions on definition of complex analytic family This is related to Kodaira's Complex Manifold and Deformation of Complex Structure Chpt 2, Sec 3(a) Complex Analytic Family. 
The book defines the complex analytic as the following. Given $\{M_b|b\in B\}$ with $M_b$ family of compact complex manifold and $B$ is a complex manifold of dimension $m$, let $(M,\omega, B)$ be triple s.t. the following conditions hold. 
(1)$\forall b\in B, \omega^{-1}(b)$ is a complex submanifold of $M$.(i.e. fibers are all submanifolds.)
(2)$\omega:M\to B$ is holomorphic of constant rank $m$ 
(3)$\forall b\in B,\omega^{-1}(b)=M_b$
I think $(1)$ is redundant. $(2)$ gives a coordinate of $M$ already. Suppose $M$'s local chart is $(z_1,\dots, z_{n-m},\dots,z_n)$ and $B$'s local chart looks like $(b_1,\dots, b_m)$. Since $\omega$ is of constant rank $m$, WLOG, we can assume $(z_1,\dots, z_{n-m+1},b_1,\dots, b_m)$ gives coordinate of local chart of $M$ by assuming $det(\frac{\partial\omega}{\partial(z_{n-m+1},\dots, z_{n})})\neq 0$ and shrinking the size of the open set. Now the newly induced $\omega$ on chart level is given by projection map. This trivially indicates $\omega^{-1}(b)$ is always a complex submanifold of $M$. 
$\textbf{Q1:}$ Is $(1)$ redundant given $(2)$?
$\textbf{Q2:}$ Is the reason to take the fiber being compact that one wants to study deformation of complex structure of compact or projectivisible manifolds? It seems that there is nothing too bad to replace compact fiber with non-compact fiber(without boundary) here. Is there something significantly wrong to look at non-compact fiber case naively?
$\textbf{Q2':}$ Since $\omega^{-1}(b)$ are all submanifold of $M$, there is no way $M_b$ degenerate(i.e. singular cases are ruled out as we demand $M_b$ compact.) How does this fit into a family with degenerate fibers? Or is it even possible?
$\textbf{Q3:}$ It seems "before taking closure", this construction only gives the open subset of moduli space. Is this correct? And this construction does not allow fiber dimension jummping here. Is this reasonable to hope? Dimension should be "semi-continuous" rather than constant in general in algebraic setting. Or have I missed the main point somewhere?
 A: Q1: Yes. (2) and (3) mean that $\omega$ is a surjective submersion, so its fibers are submanifolds.
Q2: Yes. Nothing stops you from looking at maps with noncompact fibers, but given the state of the art of noncompact complex geometry, you won't be able to say very much about that case.
Q2': This only gives you deformations of smooth manifolds (which is already a tricky enough case in complex geometry!). Kodaira's book is more or less a friendly recap of his and Spencer's papers on deformations from the late 1950's, so by reading it you follow more or less the historical development. The "real" notion of a deformation of compact complex manifolds (or algebraic varieties) is the one of a proper flat morphism $\omega : M \to B$, where $M$ and $B$ are complex (or algebraic) schemes. However, Serre only defined flat modules for the first time in 1956, at the same time Kodaira and Spencer were working out their theory, which later inspired the algebraic work on deformations. Today, we have the advantage that a lot of this theory has been cleaned up, abstracted and simplified, but at the time it wasn't clear which parts were most important.
Q3: This definition may not even give you an open set in the "moduli space". (At this point in the story, it's not even clear what that should be. Kuranishi's work comes later, and the algebraic version of representable functors also.) The definition includes trivial families, and families what would map to curves in the "moduli space", and so on. This definition is better seen as a local description of a neighborhood around a smooth point in the "moduli space" (whatever that is, if it exists). There is no dimension jumping (which I don't think there is in the algebraic case over $\mathbb C$ either, but don't hold me to that), but the dimensions of the stalks of direct images of coherent sheaves on $M$ to the base $B$ can jump in a semicontinuous way.
