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 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.
Definition
Throughout, will be some non-empty index set and for every index let be a topological space. Denote the Cartesian product of the sets by
and for every index denote the -th canonical projection by
The product topology, sometimes called the Tychonoff topology, on is defined to be the
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
form a subbase for the topology on A subset of is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form The are sometimes called
The product topology is also called the ) 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.
Examples
If the
The
Several additional examples are given in the article on the initial topology.
Properties
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
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
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:[1]
Any product of Hausdorff spaces is again a Hausdorff space.
If is fixed then the set
is a dense subset of the product space .[1]
Relation to other topological notions
Separation
- Every product of T0 spacesis 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 need not be normal.
Compactness
- Every product of compact spaces is compact (Tychonoff's theorem).
- A product of locally compact spaces need 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).
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.[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.
See also
- Disjoint union (topology) – space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology
- Final topology – Finest topology making some functions continuous
- Initial topology – Coarsest topology making certain functions continuous - Sometimes called the projective limit topology
- Inverse limit – Construction in category theory
- Pointwise convergence – A notion of convergence in mathematics
- Quotient space (topology) – Topological space construction
- Subspace (topology)– Inherited topology
- Weak topology – Mathematical term
Notes
- ^ a b Bourbaki 1989, pp. 43–50.
- ^ Pervin, William J. (1964), Foundations of General Topology, Academic Press, p. 33
- ISBN 978-0-486-65676-2
References
- OCLC 18588129.
- Willard, Stephen (1970). General Topology. Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0486434796. Retrieved 13 February 2013.