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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?

Thanks in advance!

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

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