find a base topology

Question

Exercise: Let $(X,t_1)$, $(Y,t_2)$ be topological spaces where $ X=\{1,2,3\}$, $Y=\{1,2\}$

$t_1=\{\emptyset, X,\{1\},\{2,3\} \}$,
$t_2=\{ \emptyset , Y, \{1\} \} $

find a base of the product topology $X\times Y$

my solution: we know $t_1$ is a base for $X$ and $t_2$ a base of $Y$ so we just need to find the product $t_1 \times t_2$

$t_1 \times t_2= \{\emptyset, X\times Y, \{ X\times\{1\} \},\{\{1\}\times Y\}, \{(1,1)\},\{\{2,3\}\times Y \}, \{\{2,3\}\times \{1\} \} \}$

Is the set $t_1 \times t_2$ correct ?

Answer 1

A base for $\mathcal{T}_1$ is $\{\{1\},\{2,3\}\}$ and for $\mathcal{T}_2$ it's $\{\{1\},Y\}$. (we don't need $\emptyset$ in a base).

So the generated product base is $\{\{(1,1)\}, \{(1,1), (1,2)\}, \{(2,1), (3,1)\}, \{(2,1),(2,2),(3,1),(3,2)\}\}$, so $2 \times 2 = 4$ subsets.

Write out a set like $\{2,3\} \times\{1\}$ as $\{(2,1), (3,1)\}$ etc. Show it as set of pairs.