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

will be some non-empty index set be a topological space. Denote the Cartesian product by

-th by

The , sometimes called the is defined to be the coarsest topology are continuous. It is the initial topology endowed with the product topology is called the .

is the topology generated. A subset is open if and only if it is the union of (possibly infinitely many) intersections's are sometimes called open cylinders, and their intersections are cylinder sets.

The product topology is also called the because a sequence (or more generally, a net of all real-valued functions, and convergence in the product topology is the same as pointwise convergence of functions.

Examples

If the real line is endowed with its standard topology is equal to the ordinary Euclidean topology is finite, this is also equivalent to the box topology)

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

forms a basis for what is called the box topology 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 is a continuous map, then there exists the following diagram commutes:

This shows that the product space is a product in the category of topological spaces is continuous if and only if are continuous in some way.

are open maps is a subspace) The canonical projections are not generally closed maps).

are is equal to the product of the closures:

Any product of Hausdorff spaces. This line of argument is used for showing that other topological invariants are preserved under the product topology.

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 fixed then the set

is a dense subset.

Relation to other topological notions

Separation

• Every product of T0 spacess is T0.

• Every product of T1 spacess 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 be normal. An example of this is the product of the set X of ordinals less than the first uncountable ordinal. Both spaces have the order topology is not normal.

Compactness

• Every product of compact spaces is compact (Tychonoff's theorem).

• A product of locally compact spaces be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact 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. 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.

See also

• - Sometimes called the projective limit topology

Notes

References