# The product topological space of two finite closed topologies is not finite closed.

Let $$X_1, X_2$$ be infite sets and $$\tau_1$$and $$\tau_2$$ the finite closed topologies on $$X_1$$ and $$X_2$$ respectively. Show that the product topology $$\tau$$ on $$X_1\times X_2$$ is not the finite closed topology.

As $$X_1$$ is infite then if $$\{x\}\in X$$ then $$X_1\setminus\{x\}\in\tau_1$$. Then $$\{x\}\times X_2$$ is by definition an open set in the product topology space however its complement is $$\{x\}\times X_2$$ is not finite as $$X_2$$ is infinite contradicting the fact $$\{x\}\times X_2$$ is closed. Therefore $$\{x\}\times X_2$$ is not open. Then the product topology cannot be the finite closed topology.

Questions:

Is this proof right? If not why not? Which are the alternatives?

$$\{x\} \times X_2$$ is not open, but it is closed. Since it is infinite, we are done.