I wanted to prove that $X^{\omega}$ is a compact set in product topology. $X$ is a compact space. Before entering into the Tychonoff's theorem I just wanted to examine why it is not trivial to say that product of compact spaces is compact as we did for the case of connectedness.
Can anyone please give me a non-trivial example of an open cover which will cover $X^{\omega}$ but not any obvious finite sub-cover will cover the same?