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 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 X_i, indexed by ${\displaystyle i\in I}$, and the canonical projections p_i : X → X_i, 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 p_i 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 ${\displaystyle \prod _{i\in I}U_{i}}$, where each U_i is open in X_i and U_i ≠ X_i 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 X_i gives a basis for the product topology of ${\displaystyle \prod _{i\in I}X_{i}}$. That is, for a finite product, the set of all ${\displaystyle \prod _{i\in I}U_{i}}$, where ${\displaystyle U_{i}}$ is an element of the (chosen) basis of ${\displaystyle X_{i}}$, is a basis for the product topology of ${\displaystyle \prod _{i\in I}X_{i}}$.

The product topology on X is the topology generated by sets of the form p_i⁻¹(U_i), where i is in I and U_i is an open subset of X_i. In other words, the sets {p_i⁻¹(U_i)} 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 p_i⁻¹(U_i). The p_i⁻¹(U_i) are sometimes called open cylinders, and their intersections are cylinder sets.

In general, the product of the topologies of each X_i 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, f_i : Y → X_i 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 f_i = p_i ∘ f is continuous for all i in I. In many cases it is easier to check that the component functions f_i are continuous. Checking whether a map f : Y → X is continuous is usually more difficult; one tries to use the fact that the p_i are continuous in some way.

In addition to being continuous, the canonical projections p_i : X → X_i are open maps. This means that any open subset of the product space remains open when projected down to the X_i. The converse is not true: if W is a subspace of the product space whose projections down to all the X_i 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 ${\displaystyle \{(x,y)\in \mathbb {R} ^{2}\mid xy=1\},}$ 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 X_i converge. In particular, if one considers the space X = R^I 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 X_i 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

1. "Product topology preserves the Hausdorff property". PlanetMath.
2. Pervin, William J. (1964), Foundations of General Topology, Academic Press, p. 33
3. 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.