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I am aware of the result that if you take the cartesian product of any collection of compact sets, whether it be finite or infinite (by Tychonoff's theorem.)

My question is, suppose we have a set (let's call it $\mathcal{A}=\lbrace A_i\rbrace_{i\in I}$) of compact sets. If we define $$C=\underset{i\in I}{\Pi}A_i$$ is $C$ conmmpact. Intuitively, I would say yes, it is compact, however my TA disagreed with me and now I am here.

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    It's not exactly clear to me what space you're considering. For example, if the compact sets are $X_a$ for $a \in A$, could you write the product? – 2017-02-12
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    Suppose A = N. Suppose each of the compact sets X_n is of the form [n,n+1]. Then, is the product X_1 × X_2 × ... compact? – 2017-02-13
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    Yes, but this example obviously satisfies the hypotheses of Tychonoff's Theorem. – 2017-02-13

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As user414998 said this is exactly the hypothesis of Tychonoff's theorem so the answer is yes, $C$ is compact.

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    It is compact only if you put on $C$ the product topology, of course. – 2017-02-16