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.
Denote the Cartesian product of the sets
is defined to be the coarsest topology (that is, the topology with the fewest open sets) for which all the projections
The open sets in the product topology are arbitrary unions (finite or infinite) of sets of the form
is the topology generated by sets of the form
is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form
The product topology is also called the topology of pointwise convergence because a sequence (or more generally, a net) in
is equal to the ordinary Euclidean topology on
is finite, this is also equivalent to the box topology on
) The Cantor set is homeomorphic to the product of countably many copies of the discrete space
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.
forms a basis for what is called the box topology on
In general, the box topology is finer than the product topology, but for finite products they coincide.
together with the canonical projections, can be characterized by the following universal property: if
the following diagram commutes: This shows that the product space is a product in the category of topological spaces.
In many cases it is easier to check that the component functions
In addition to being continuous, the canonical projections
This means that any open subset of the product space remains open when projected down to the
is a subspace of the product space whose projections down to all the
) The canonical projections are not generally closed maps (consider for example the closed set
is a closed subset of the product space
of arbitrary subsets in the product 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.
is a dense subset of the product space
[1] Separation Compactness Connectedness Metric spaces 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.
[2] 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,[3] and shows why the product topology may be considered the more useful topology to put on a Cartesian product.