Let $X$ and $Y$ are topological spaces with indiscrete topologies then prove that the product topology $X\times Y$ will be indiscrete space

Question

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

Answer 1

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

Answer 2

Hint: What's $\emptyset\times Y$? Also, beware that generally (for finite products) you don't have just products of open sets, but unions of products of open sets as well.

Answer 3

$$(x,y)\in \varnothing\times Y\iff x\in\varnothing\wedge y\in Y$$

Can you find a pair $(x,y)\in X\times Y$ having this property?

What do you conclude from that?