Let $X=\mathbb{R}$ with the lower limit topology, and $Y=\mathbb{R}$ with the upper limit topology. Is $[1,2) \times [1,2)$ open in $X \times Y$

Question

I don't believe $[1,2)$ is open in Y, so the product topology, $X\times Y$, is then not open.

As I'm reading Topology Without Tears, I see Proposition 8.1.4 that discusses the product space being closed if the subsets of the topological spaces are closed, but nothing about when mixed.

Answer 1

Because projections are open maps, $A \times B$ open in $X \times Y$ implies that both $A = \pi_X[A \times B]$ is open in $X$ and $B = \pi_Y[A \times B]$ is open in $Y$. As $B = [1,2)$ is not open in the upper limit topology (generated by sets of the form $(x,y], x < y$) (as $1$ is not an interior point of $B$), $A \times B$ cannot be open in the product.