subsets in the product space is equal to the product of the closures: Any product of Hausdorff spaces is again a Hausdorff space. Tychonoff's theorem, which is equivalent to the axiom of choice, states that any product of compact spaces is a compact space. A specialization of Tychonoff's theorem that requires only the ultrafilter lemma (and not the full strength of the axiom of choice) states that any product of compact Hausdorff spaces is a compact space. If is fixed then the set is a dense subset of the product space . Relation to other topological notions Separation Every product of T0 spaces is T0. Every product of T1 spaces is T1. Every product of Hausdorff spaces is Hausdorff. Every product of regular spaces is regular. Every product of Tychonoff spaces is Tychonoff. A product of normal spaces be normal. Compactness Every product of compact spaces is compact (Tychonoff's theorem). A product of locally compact spaces be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact locally compact (This condition is sufficient and necessary). Connectedness Every product of connected (resp. path-connected) spaces is connected (resp. path-connected). Every product of hereditarily disconnected spaces is hereditarily disconnected. Metric spaces Countable products of metric spaces are metrizable spaces. Axiom of choice One of many ways to express the axiom of choice is to say that it is equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. The proof that this is equivalent to the statement of the axiom in terms of choice functions is immediate: one needs only to pick an element from each set to find a representative in the product. Conversely, a representative of the product is a set which contains exactly one element from each component. The axiom of choice occurs again in the study of (topological) product spaces; for example, Tychonoff's theorem on compact sets is a more complex and subtle example of a statement that requires the axiom of choice and is equivalent to it in its most general formulation, and shows why the product topology may be considered the more useful topology to put on a Cartesian product. See also - Sometimes called the projective limit topology Notes References General topology Operations on structures