Property of a morphism inherited by the fibers Let $T$ be a scheme over an algebraically closed field $k$ and let $f:X\to Y$ be a $k$-morphism over $T$. 
For which properties P is it true that if $f_t:X_t\to Y_t$ has P for all geometric points $t\in T(k)$, then $f$ has P? Is it true for open immersions, for instance? What about properness, flatness...? (Feel free to add some assumptions, I really have no idea whether I am asking something very strong or not!)
 A: Suppose $X,Y$ are flat and of finite presentation over $T$.
Then, let $P$ be a property in $\{$flat, smooth, etale, open immersion, isomorphism, flat and a relative complete intersection morphism$\}$, then $f$ has property $P$ if and only if each $f_t$ has $P$ for all geometric points $t$ in $T$ (Here $T$ doesn't have to be a $k$-scheme).
This is stated in Deligne/Rapoport on page "DeRa-28" (page number 170), section I,(7.4), and he gives references to EGA IV 11.3.10 and 17.9.1. A very special case for $P = $ smooth for curves over Dedekind domains is given in Silverman's Advanced Topics in Arithmetic of Elliptic Curves, in the chapter on Neron models, section 2 (or maybe 3, or 4?)
A: No, open immersion of fibers does not imply actual open immersion.  
For example take $T=\mathbb A^1_k, X=\mathbb A^1_k, Y=\mathbb A^2_k$
For the structural morphisms  $X\to T$ take the constant map  $x\mapsto 0$ and for $Y\to T$ take the first projection.
Finally for $X\to Y$ take $x\mapsto (x,0)$, which is not an open embeding but is a fiberwise open embedding.
