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Product topology
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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 pullback of its factors, whereas the box topology is too fine; this is the sense in which the product topology is "natural".
Definition
Let I be a (possibly infinite) index set and suppose Xi is a topological space for every i in I. Set X = Π Xi, the Cartesian product of the sets Xi. For every i in I, we have a canonical projection 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.
Explicitly, the product topology on X can be described as the topology generated by sets of the form pi−1(U), where i in I and U is an open subset of Xi. In other words, the sets form a subbase for the topology on X. A subset of X is open if and only if it is a union of (possibly infinitely many) intersections of finitely many sets of the form pi−1(U). The pi−1(U) are sometimes called open cylinders, and their intersections are cylinder sets.
We can describe a basis for the product topology using bases of the constituting spaces Xi. A basis consists of sets , where for cofinitely many (all but finitely many) i, (it's the whole space), and otherwise it's a basic open set of .
In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the Xi gives a basis for the product .
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 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:
This shows that the product space is a product in the category of topological spaces. If follows from the above universal property that a map f : Y → X is continuous iff fi = pi o f is continuous for all i in I. In many cases it is often easier to check that the component functions fi are continuous. Checking whether a map g : X→ Z 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 whose projections onto both axes are R \ ).
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
- Compactness
- Every product of compact spaces is compact
- A product of locally compact spaces need not be locally compact
- Connectedness
- Every product of connected (resp. path-connected) spaces is connected (resp. path-connected)
- Every product of hereditarily disconnected spaces is hereditarily disconnected.
A map that "locally looks like" a canonical projection F × U → U is called a fiber bundle.
Axiom of choiceThe axiom of choice is equivalent to the statement that the product of a non-empty collection of non-empty sets is non-empty. The proof is easy enough: 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 more generally in 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.
See also
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