$X = \{a,b,c,d\}$ is given the topology $T_X = \{ \emptyset, \{a\}, \{a,c,d\}, \{c,d\}, X \}$ and $Y = \{1,2,3\}$ given the topology $T_Y = \{ \emptyset, \{1\}, \{1,3\}, Y \}$.
Will a basis for the product topology on $X \times Y$ be:
$B_{X \times Y} = \{ \{a\} \times \{1\}, \{a\} \times \{1,3\}, \{a\} \times Y, \{c,d\} \times \{1\}, \{c,d\}\times \{1,3\}, \{c,d\}\times Y \}$
or
$B_{XxY} = \{ \{a\} \times \{1\}, \{a,c,d\} \times \{1\}, \{c,d\} \times\{1\}, \{a\} \times \{1,3\}, \{a,c,d\} \times \{1,3\}, \{c,d\} \times \{1,3\}, \{a\} \times Y, \{c,d\} \times Y, \{c,d\} \times Y \}$
and if so; why is $B_{XxY} = T_{XxY}$ and the union of $\{a\}$ and $\{c,d\}$ included in the basis?
Also, should the empty set be included in the basis?