Recently, I'm considerring an open problem from the paper by Ofelia T.Alas and others':On the extent of star countable spaces. It easily can be downloaded by google. The open Problem is this:
Suppose that $X$ is a (strongly) monotonically monolithic star countable space. Must $X$ be Lindelof?
Somebody has proved it is true that if $X$ is a strongly monotonically monolithic star countable space, then $X$ must be Lindelof. (see the paper: A note on the extent of two subclasses of star countable spaces by the author Zuoming Yu).
In the morning, I met acrossly an example See Dan Ma's discussion non-normal product space: there exist a separable metric space $X$ and a Lindelöf space $Y$, such that $X \times Y$ is not a Lindelöf space.
On one hand because every point-countable base space is strongly monotonically monolithic, according to Tkachuk, "monolithic spaces and D-spaces revisited", Proposition 2.5, then $X$ of course is strongly monotonically monolithic. And $Y$ is also have point-countable base, so $Y$ is also strongly monotonically monolithic. Moreover, $X\times Y$ is strongly monotonically monolithic (see the same paper of Tkachuk: theorem 2.10).
On the other hand, The space $X$ and the space $Y$ are two star countable, clearly, their product is also star countable.
Now the contradiction comes, such space $X \times Y$ is not normal, and hence is not Lindelof, which contradicts with the result of Zuoming Yu.
I don't know what's wrong. Could somebody help? Thanks for any help.
Added: The product of 2 star-countable spaces is star-countable:
Suppose $X$ and $Y$ are both star countable. Given any open cover of basic nbhd $U\times V=\{u\times v: u \in U; v\in V\}$ of $X\times Y$. Clearly, $U$ is an open cover of $X$ and $V$ is an open cover of $Y$, therefore, there exist countable subsets $C\subset X$ and $D\subset Y$ which satisfies that $St(C,U)=X$ and $St(D,U)=Y$. Now we can get an countable subset: $C\times D$ of $X\times Y$ and easily see that $St(C\times D, U\times V)=X\times Y$.
Do you have a reference or proof for "the product of 2 star-countable spaces is star-countable"? I don't see that. And your $X$ and $Y$ (the Michael line but based on a Bernstein set.. as $Y$) could be a counterexample.
As to your (added) proof: pick $(x,y) \in X \times Y$. We get independent $U$ and $V$ for the different coordinates, but we can not yet say that the product of these is in the originla cover of product sets!