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
Given X, also known as the product space, such that
\( {\displaystyle X:=\prod _{i\in I}X_{i}} \_
is the Cartesian product of the topological spaces Xi, indexed by \( i\in I, \) and the canonical projections pi : X → Xi, the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonoff topology.
The open sets in the product topology are unions (finite or infinite) of sets of the form \( \prod _{i\in I}U_{i} \) , where each Ui is open in Xi and Ui ≠ Xi for only finitely many i. 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 Xi gives a basis for the product topology of ∏ i ∈ I X i {\displaystyle \prod _{i\in I}X_{i}} \prod _{i\in I}X_{i}. That is, for a finite product, the set of all \( \prod _{i\in I}U_{i} \), where \( U_{i} \) is an element of the (chosen) basis of \( X_{i} \) , is a basis for the product topology of \( \prod _{i\in I}X_{i} \).
The product topology on X is the topology generated by sets of the form pi−1(Ui), where i is in I and Ui is an open subset of Xi. In other words, the sets {pi−1(Ui)} form a subbase for the topology on X. A subset of X is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form pi−1(Ui). The pi−1(Ui) are sometimes called open cylinders, and their intersections are cylinder sets.
In general, the product of the topologies of each Xi forms a basis for what is called the box topology on X. In general, the box topology is finer than the product topology, but for finite products they coincide.
Examples
If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on Rn.
The Cantor set is homeomorphic to the product of countably many copies of the discrete space {0,1} 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 X, together with the canonical projections, can be characterized by the following universal property: If Y is a topological space, and for every i in I, fi : Y → Xi is a continuous map, then there exists precisely one continuous map f : Y → X such that for each i in I the following diagram commutes:
Characteristic property of product spaces
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 f : Y → X is continuous if and only if fi = pi ∘ f is continuous for all i in I. In many cases it is easier to check that the component functions fi are continuous. Checking whether a map f : Y → X is continuous is usually more difficult; one tries to use the fact that the pi are continuous in some way.
In addition to being continuous, the canonical projections pi : X → Xi are open maps. This means that any open subset of the product space remains open when projected down to the Xi. The converse is not true: if W is a subspace of the product space whose projections down to all the Xi are open, then W need not be open in X. (Consider for instance W = R2 \ (0,1)2.) The canonical projections are not generally closed maps (consider for example the closed set \( \{(x,y)\in {\mathbb {R}}^{2}\mid xy=1\} \), whose projections onto both axes are R \ {0}).
The product topology is also called the topology of pointwise convergence because of the following fact: a sequence (or net) in X converges if and only if all its projections to the spaces Xi converge. In particular, if one considers the space X = RI of all real valued functions on I, convergence in the product topology is the same as pointwise convergence of functions.
Any product of closed subsets of Xi is a closed set in X.
An important theorem about the product topology is Tychonoff's theorem: any product of compact spaces is compact. This is easy to show for finite products, while the general statement is equivalent to the axiom of choice.
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[1]
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
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 is equivalent to the axiom of choice,[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)
Projective limit topology
Quotient space
Subspace (topology)
Notes
"Product topology preserves the Hausdorff property". PlanetMath.
Pervin, William J. (1964), Foundations of General Topology, Academic Press, p. 33
Hocking, John G.; Young, Gail S. (1988) [1961], Topology, Dover, p. 28, ISBN 978-0-486-65676-2
References
Willard, Stephen (1970). General Topology. Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0486434796. Retrieved 13 February 2013.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
