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 

It turns out that $X\times\emptyset=\emptyset\times Y=\emptyset$. So, yes, the product topology on $X\times Y$ is the indiscrete topology.
