Suppose $\kappa$ is an inaccessible cardinal, and let $P$ be the $< \aleph\_1-$support product of $Add(\alpha^{++}, 1)$ for singular cardinals $\alpha < \kappa.$ 


1- Does this forcing preserve cardinals?


2-(A weaker question) Does $\kappa$ remain inaccessible in the generic extension? 



$< \aleph_1-$support Product of Cohen forcings
Your forcing notion will collapse all uncountable cardinals
below $\kappa$ to $\aleph\_1$. To see this, fix any
uncountable $\gamma\lt\kappa$. Consider the first
$\aleph\_1$ many cardinals after $\gamma$ at which forcing
occurs. For the $\xi^{th}$ such cardinal $\alpha$, we are adding a
subset to $\alpha^{++}$. Consider the first ordinal in the set
added at this stage. This can be any ordinal up to
$\alpha^{++}$, including any ordinal below $\gamma$. Because you are using countable support, any condition specifies nontrivial sets only on a countable number of these cardinals, and so it is dense that any particular $\beta\lt\gamma$ appears at least
once in such a way. Thus, in the generic extension, there
will be a surjective map from $\aleph\_1$ onto $\gamma$, and
so $\gamma$ is collapsed.


The cardinal $\kappa$ itself is not collapsed, by a
$\Delta$-system argument (and neither is any cardinal above
$\kappa$ collapsed), and so this forcing makes $\kappa$ the
$\aleph\_2$ of the extension. In particular, it does not
preserve the inaccessibility of $\kappa$.


