Proving finite product of discrete topology is discrete

Question

Exercise: If $(X,\tau_1)$,$(X,\tau_2)$....$(X\tau_n)$ are discrete spaces. Prove that the product space $(X,\tau_1)\times(X,\tau_2)....\times(X,\tau_n)$ is also a discrete space.

I had a very straightforward idea that I do not know if it is right. $\{a_1,a_2,...a_n,\:a_i\in\tau_i,i=1,2...n\}$ forms a basis for the topology $\tau_n$ that can generate any element then the topology $\tau_n$ must be discrete.

Question:

Is my proof right? What are other alternative proofs?

Thanks in advance!

Answer 1

Your proof makes no sense, because the set that you defined makes no sense.

You can prove it as follows: is $p_1\in X_1$, $p_2\in X_2$, …, $p_n\in X_n$, then each set $\{p_i\}$ is open and therefore, $\bigl\{(p_1,p_2,\ldots,p_n)\bigr\}$ is an open subset of $\prod_{i=1}^nX_i$, since it is equal to $\{p_1\}\times\{p_2\}\times\cdots\times\{p_n\}$. Since each singleton is open, your topology is discrete.