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An arbitrary product of compact spaces is compact in the product topology. This deep result is equivalent to the Axiom of Choice. Compactness is stable under infinite products.
From your study of compact Hausdorff spaces, you know that compactness — every open cover has a finite subcover — is one of the most powerful properties a topological space can have. It generalizes "closed and bounded" from ℝⁿ to abstract spaces and underlies the most important theorems in analysis: continuous functions on compact spaces attain their maximum, are uniformly continuous, and map compact sets to compact sets. From your study of the product topology, you know how to form the product ∏Xα of a family of topological spaces, with open sets generated by boxes that constrain only finitely many coordinates. The question Tychonoff's theorem answers is: if each factor Xα is compact, is the product compact?
The answer is yes, and this is far from obvious. A product of infinitely many spaces — even infinitely many copies of [0, 1] — is a very large, intricate space. An open cover of ∏Xα involves open sets that can constrain different coordinates in different ways, and there is no finite set of coordinates you can use to describe the whole product. The key insight is that the product topology is the *coarsest* topology making all projections πα: ∏Xβ → Xα continuous. Its open sets are not full boxes but merely finite intersections of preimages of open sets in the factors. This sparseness in the topology is exactly what makes compactness survive.
The standard proof of Tychonoff's theorem uses either Zorn's lemma or the ultrafilter lemma, both equivalent to the Axiom of Choice. The ultrafilter proof is elegant: an ultrafilter on a compact space always converges, and an ultrafilter on a product space can be projected to each factor to get an ultrafilter on Xα, which converges by the compactness of Xα. The projections converge to a point in each Xα, and by the definition of the product topology, the ultrafilter on the product converges to the corresponding point in ∏Xα. The theorem is in fact *equivalent* to the Axiom of Choice: Kelley proved in 1950 that if you assume Tychonoff for all spaces (not just Hausdorff ones), you can derive the Axiom of Choice.
Tychonoff's theorem is foundational for functional analysis and the study of infinite-dimensional spaces. The Banach–Alaoglu theorem — that the closed unit ball of the dual of a Banach space is compact in the weak-* topology — is a direct consequence. The Stone–Čech compactification, your next topic, uses Tychonoff's theorem to embed any completely regular space into a compact Hausdorff space by constructing it as a product of copies of [0, 1]. These applications illustrate why infinite products matter: they provide a universal construction for building compact spaces out of simpler pieces.