Let $X$ and $Y$ are topological spaces with indiscrete topologies then prove that the product topology $X\times Y$ will be indiscrete space Now $\tau_X=\{X,\emptyset\}$ and $\tau_Y=\{Y,\emptyset\}$ their product topology will be like $\tau_{X\times Y}=\{X \times Y , \emptyset \times Y , X \times \emptyset , \emptyset\}$ which is clearly not indiscrete . Please help where I am going wrong. Thanks in advance 

$$(x,y)\in \varnothing\times Y\iff x\in\varnothing\wedge y\in Y$$
Can you find a pair $(x,y)\in X\times Y$ having this property?
What do you conclude from that?
