Tychonoff's theorem

Tychonoff's theorem states that the product of any family of compact topological spaces is compact when given the product topology. It applies even to infinitely many spaces.

Product topology: The coarsest topology on a product set that makes every coordinate projection continuous. It is the topology in which Tychonoff’s product remains compact.
Open cover: A collection of open sets whose union contains the space under consideration. Finite subcovers define compactness in the theorem’s hypotheses and conclusion.
Andrey Tychonoff: A Soviet mathematician whose work included foundational contributions to general topology. His name is attached to the product compactness theorem.
Banach–Alaoglu theorem: A theorem that the closed unit ball of a dual normed space is compact in the weak-* topology. Its proof uses product compactness to establish compactness of dual-space balls.
Heine–Borel theorem: A theorem characterizing compact subsets of Euclidean space as closed and bounded. It handles finite-dimensional Euclidean compactness, unlike arbitrary topological products.
Compact space: A topological space in which every open cover has a finite subcover. Compactness of each factor is the hypothesis preserved by the product.
Cartesian product: The set of tuples formed by choosing one element from each member of a family of sets. The theorem concerns the topology placed on this underlying set.
General topology: The branch of topology studying topological spaces and continuous maps in broad settings. Tychonoff’s theorem became a central structural result in this field.
Weak-* topology: The topology on a dual space defined by pointwise convergence on the original space. It embeds dual balls into products, where Tychonoff’s theorem applies.
Sequential compactness: A property requiring every sequence to have a convergent subsequence. It often differs from compactness in general spaces, so it cannot replace the theorem’s conclusion.

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