Properties of product topology I have example of property like this:
"Let $\{X_\lambda, \lambda \in \Lambda\}$ be any family of topological spaces and $\forall\lambda \in \Lambda$,  $A_\lambda\subseteq X_\lambda$. $\prod_{\lambda \in \Lambda }A_\lambda$ is dense in $\prod_{\lambda \in \Lambda }X_\lambda$ iff $A_\lambda$ is dense in $X_{\lambda}$ for every $\lambda \in \Lambda$"
I proved this and now I have to check in what condition (and if) same statement is true for these properties (instead of density)
1. Separability
2. first-countable
3. second-countable
4. metrizability  
I know how to prove 4. and it is only true for countable family.
What about others? And maybe some short idea for proof or example it isn't true?
 A: The following are the best we can do:
1) a product of $\mathfrak{c}$ (size of the reals) separable spaces is separable, but a larger product is no longer separable (unless we add trivial spaces like singletons).
2) a product of countably many first countable spaces is first countable (and larger products won't be, modulo trivialities).
3) a product of countably many second countable spaces is second countable (and not for larger products etc.)
4) a product of countably many metrizable spaces is metrizable (and not for larger products etc.)
The space $\{0,1\}^{I}$ where $I$ is any uncountable set is an example of a non-first countable (so non-metrizable, non-second countable) product, and if we pick $|I| > |\mathbb{R}|$ we have a non-separable product of separable spaces.
So all these properties need some restrictions on the number of (non-trivial) factors to make them being preserved by products. This contrasts with properties like (path-)connectedness and compactness which are preserved by all products, but makes them much better behaved than properties like normality or paracompactness which can be lost when we take the product  of two spaces already.
