How to find subbase and base for $X\times Y$?

Question

Let $\tau :=\{X,\emptyset,\{a\},\{b,c\}\} $ on $X=\{a,b,c\}$ and $\tau^*:=\{Y,\emptyset,\{u\}\}$ on $Y:=\{u,v\}$

i) Find a subbase for the product topology on $X\times Y$

ii) Find a base for the product topology on $X\times Y$

I found the product topology as $\tau^1=\{X\times Y,\emptyset,X\times \{u\},\{a\}\times Y,\{a\}\times \{u\},\{b,c\}\times Y,\{b,c\}\times \{u\}\}$

Is this correct? How can I continue?

Answer 1

Unless I'm overlooking something here, what you have there is a base (and hence a subbase) for the product topology, but not a topology in itself.

Remember: the product topology for $X\times Y$ is defined as the topology generated by the base $\{U\times V \mid U\in \tau, V\in \tau^*\}$. In general, this base is not a topology, as is the case here.

Note that the union of two open sets should be open. In the $\tau^1$ you provided, $(\{a\}\times \{u\})\cup (\{b\}\times Y)=\{(a,u), (b,u), (b,v)\}$ is nowhere to be found.