Consider the topological space $ X = (\mathbb{R}_l \times \mathbb{R}, \tau_{prod})$. Here $\mathbb{R}_l$ denotes the real line with the lower limit topology. The basis for the product space is the collection $$ \beta = \left\{ [a, b) \times (c,d) \mid a, b, c, d \in \mathbb{R} \right\}. $$ Fix some $q \in \mathbb{Q}$. I want to consider the set $A = \left\{q \right\} \times \mathbb{Q}$. In the plane this is just like a vertical line extending onwards from $q$, and consisting of all rationals.
I can write $$A = \bigcup_{q' \in \mathbb{Q}} \left\{q \times q' \right\}. $$ I was now trying to figure out what is the closure of the singleton $\left\{q \times q' \right\}$ in $X$ for the product topology. I think this set is closed, so the closure equals itself. I wasn't sure how to prove this. I let $z_1 \times z_2$ be a point in the closure. Then for any basis open $V = [a,b) \times (c,d)$ containing $z_1 \times z_2$ we have that $V \cap \left\{q \times q' \right\} \neq \emptyset. $ I want to conclude from this that $z_1 \times z_2 \in \left\{q \times q'\right\}$, but not sure how to.
Also, if my intuition is correct, is the interior of $\left\{q \times q' \right\}$ empty? The reason I'm asking this, is because I want to show that the set $\mathbb{Q} \times \mathbb{Q}$ is meagre (i.e. can be written as a countable union of nowhere dense sets) in $X$. Because then I could write $\mathbb{Q} \times \mathbb{Q}$ as a countable union of sets of the form $A$, from which the conclusion would follow.
Thank you in advance for any help.
The topology $\tau_l$ of the space $\Bbb R_l$, usually called the Sorgenfrey line or arrow, is related with the standard topology $\tau$ of $\Bbb R$. Namely, $\tau_l$ is stronger than $\tau$. In particular, this implies that all singletons in spaces $\Bbb R_l$ and $\Bbb R_l\times \Bbb R$ are closed. On the other hand, each open non-empty open subset of the space $\Bbb R_l$ contains a non-empty open subset of the space $\Bbb R$. This easily implies that the spaces $\Bbb R_l$ and $\Bbb R$ have the same nowhere dense subsets and so the same meager subsets. The same concerns the spaces $\Bbb R_l\times \Bbb R $ and $\Bbb R\times \Bbb R $. In particular, the set $\Bbb Q\times\Bbb Q$ is meager in $\Bbb R_l\times \Bbb R $.