Topology on Cartesian products of topological spaces
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 natural-seeming, 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.
The product topology, sometimes called the Tychonoff topology, on is defined to be the coarsest topology (that is, the topology with the fewest open sets) for which all the projections are continuous. The Cartesian product endowed with the product topology is called the product space.
The open sets in the product topology are arbitrary unions (finite or infinite) of sets of the form where each is open in and for only finitely many 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 gives a basis for the product topology of That is, for a finite product, the set of all where is an element of the (chosen) basis of is a basis for the product topology of
The product topology on is the topology generated by sets of the form where and is an open subset of In other words, the sets
The product topology is also called the topology of pointwise convergence because a sequence (or more generally, a net) in converges if and only if all its projections to the spaces converge.
Explicitly, a sequence (respectively, a net ) converges to a given point if and only if in for every index where denotes (respectively, denotes ).
In particular, if is used for all then the Cartesian product is the space of all real-valued functions on and convergence in the product topology is the same as pointwise convergence of functions.
The set of Cartesian products between the open sets of the topologies of each 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.
The product space together with the canonical projections, can be characterized by the following universal property: if is a topological space, and for every is a continuous map, then there exists precisely one continuous map such that for each 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 is continuous if and only if is continuous for all In many cases it is easier to check that the component functions are continuous. Checking whether a map is continuous is usually more difficult; one tries to use the fact that the are continuous in some way.
In addition to being continuous, the canonical projections are open maps. This means that any open subset of the product space remains open when projected down to the The converse is not true: if is a subspace of the product space whose projections down to all the are open, then need not be open in (consider for instance ) The canonical projections are not generally closed maps (consider for example the closed set whose projections onto both axes are ).
Suppose is a product of arbitrary subsets, where for every If all are non-empty then is a closed subset of the product space if and only if every is a closed subset of More generally, the closure of the product of arbitrary subsets in the product space is equal to the product of the closures:
A product of locally compact spacesneed not be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact is locally compact (This condition is sufficient and necessary).
Every product of connected (resp. path-connected) spaces is connected (resp. path-connected).
Every product of hereditarily disconnected spaces is hereditarily disconnected.
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.