Product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product  topology is the natural topology on the Cartesian product. Definition. Throughout, formula_1 will be some non-empty index set and for every index formula_2 formula_3 will be a topological space. Let be the Cartesian product of the sets formula_5 and denote the ' by formula_6 The ', sometimes called the ', on formula_7 is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections formula_8 are continuous. The Cartesian product formula_4 endowed with the product topology is called the '. The product topology is also called the because of the following fact: a sequence (or  net) in formula_7 converges if and only if all its projections to the spaces formula_3 converge. In particular, if one considers the space formula_12 of all real valued functions on formula_13 convergence in the product topology is the same as pointwise convergence of functions. The open sets in the product topology are unions (finite or infinite) of sets of the form formula_14 where each formula_15 is open in formula_3 and formula_17 for only finitely many formula_18 In particular, for a finite product (in particular, for the product of two topological spaces), the set of all Cartesian products between one basis  element from each formula_3 gives a basis for the product topology of formula_20 That is, for a finite product, the set of all formula_14 where formula_15 is an element of the (chosen) basis of formula_5 is a basis for the product topology of formula_20 The product topology on formula_7 is the topology generated by sets of the form formula_26 where formula_27 and formula_15 is an open subset of formula_29 In other words, the sets form a subbase for the topology on formula_31 A subset of formula_7 is open if and only if it is a (possibly infinite) union of intersections  of finitely many sets of the form formula_33 The formula_34 are sometimes called open cylinders, and their intersections are cylinder sets. The set of Cartesian products between the open sets of the topologies of each formula_3 forms a basis for what is called the box topology on formula_31 In general, the box topology is finer than the product topology, but for finite products they coincide. Examples. If the real line formula_37 is endowed with its standard topology then the product topology on the product of formula_38 copies of formula_37 is equal to the ordinary Euclidean topology on formula_40 The Cantor  set is homeomorphic to the product of countably many copies of the discrete space formula_41 and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology. Several additional examples are given in the article on the initial topology. Properties. The product space formula_42 together with the canonical projections, can be characterized by the following universal property: if formula_43 is a topological space, and for every formula_2 formula_45 is a continuous map, then there exists continuous map formula_46 such that for each formula_27 the following  diagram commutes. This shows that the product space is a product in the category of topological spaces. It follows from the above universal property that a map formula_46 is continuous if and only if formula_49 is continuous for all formula_50 In many cases it is easier to check that the component functions formula_51 are continuous. Checking whether a map formula_52 is continuous is usually more difficult; one tries to use the fact that the formula_8 are continuous in some way. In addition to being continuous, the canonical projections formula_54 are open maps. This means that any open subset of the  product space remains open when projected down to the formula_29 The converse is not true: if formula_56 is a subspace of the product space whose projections down to all the formula_3 are open, then formula_56 need not be open in formula_7 (consider for instance formula_60) The canonical projections are not generally closed maps (consider for example the closed set formula_61 whose projections onto both axes are formula_62). Suppose formula_63 is a product of arbitrary subsets, where formula_64 for every formula_50 If all formula_66 are then formula_63 is a closed subset of the product space formula_7 if and only if every  formula_66 is a closed subset of formula_29 More generally, the closure of the product formula_63 of arbitrary subsets in the product space formula_7 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 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 formula_74 is fixed  then the set is a dense subset of the product space formula_7. 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.