Product topology

In the mathematical branch of topology, the product topology is the most "natural " topology that makes a Cartesian product of topological spaces themselves to a topological space.

Definition

For each of a ( possibly infinite ) index set is a topological space. Be the Cartesian product of the sets. For each index denote the canonical projection. Then the product topology is defined as the coarsest topology on ( the topology with the fewest open sets ) of all projections are continuous with respect. This is called the product space equipped with this topology.

Explicit description

One can describe the topology explicitly. The inverse images of open sets of the factor spaces under the canonical projections form a sub-base of the product topology, ie a subset of is open if and only if they (possibly infinitely many ) averages is the union of finitely many sets of the form, where is in and is an open subset of. It follows that, in general, not all cartesian products of open subsets must be open ( is finite, then it is, however, always so ).

Universal property

The product space together with the canonical projections is characterized by the following universal property: If a topological space and for each is continuous, then there exists a continuous function such that for all. Thus, the Cartesian product with the product topology is the product in the category of topological spaces.

Examples

  • If two metric spaces are, then you get the product topology on the metric
  • The product topology on the fold Cartesian product of the real numbers is the usual Euclidean topology.
  • The product topology on a function space is the topology of pointwise convergence.
  • The Cantor set is homeomorphic to the product space of countably many copies of the discrete space {0, 1}.
  • The space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers with the discrete topology.
  • The ring around the p- adic numbers is provided with the product topology of the discrete spaces and is then compact. This topology is also generated by the p- adic amount to.

Properties

The product topology is also called topology of pointwise convergence due to the following property: A sequence in converges if and only if all projections converge on. In particular, for the space of all functions of the convergence according to the product topology equivalent to the convergence point by point.

To check whether a given function is continuous, one can use the following criterion: is continuous if and only if all are continuous. The test for whether a function is continuous, is usually more difficult; you then try somehow the continuity of the exploit.

An important theorem about the product topology is the set of Tikhonov: Any product of compact spaces is compact. This is easy to show for finite products, but the statement is also surprisingly true for infinite products, whose proof is then but requires the axiom of choice.

Major parts of the theory of product topology were developed by AN Tikhonov.

Others

  • A related term is the sum topology.
  • The product topology is a special initial topology.
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