Product topology

In the mathematical sub-area of topology , the product topology is the “most natural” topology that turns a Cartesian product of topological spaces into a topological space itself.

definition

For each of a (possibly infinite) index set, let it be a topological space. Let be the Cartesian product of the sets . For each index denote the canonical projection . Then the product topology is defined as the coarsest topology (the topology with the fewest open sets) with respect to which all projections are continuous . With this topology one calls the product space the . ${\ displaystyle i}$ ${\ displaystyle I}$ ${\ displaystyle X_ {i}}$ ${\ displaystyle X = \ textstyle \ prod _ {i \ in I} X_ {i}}$ ${\ displaystyle X_ {i}}$ ${\ displaystyle i \ in I}$ ${\ displaystyle p_ {i} \ colon X \ to X_ {i}}$ ${\ displaystyle X}$ ${\ displaystyle p_ {i}}$ ${\ displaystyle X}$ ${\ displaystyle X_ {i}}$

Explicit description

The topology can be described explicitly. The archetypes of open sets of the factor spaces under the canonical projections form a sub-basis of the product topology, i. H. a subset is open if and only if it is the union of (possibly infinitely many) sets , each of which can be represented as finite averages of sets . Where lies in and are open subsets of . It does not follow from this that in general all Cartesian products of open subsets must be open. This only applies when is finite. ${\ displaystyle X}$ ${\ displaystyle X_ {i}}$ ${\ displaystyle p_ {i} \ colon X \ to X_ {i}}$ ${\ displaystyle Y \ subset X}$ ${\ displaystyle Y ^ {(\ alpha)}}$ ${\ displaystyle Y_ {i, k} ^ {(\ alpha)}: = p_ {i} ^ {- 1} (Y_ {k} ^ {(\ alpha)})}$ ${\ displaystyle i}$ ${\ displaystyle I}$ ${\ displaystyle Y_ {k} ^ {(\ alpha)}}$ ${\ displaystyle X_ {i}}$ ${\ displaystyle I}$

Universal property

The product space together with the canonical projections is characterized by the following universal property : If is a topological space and is continuous for each , then there is exactly one continuous function , so that applies to all . The Cartesian product with the product topology is thus the product in the category of topological spaces. ${\ displaystyle X}$ ${\ displaystyle p_ {i}}$ ${\ displaystyle Y}$ ${\ displaystyle i \ in I}$ ${\ displaystyle f_ {i} \ colon Y \ to X_ {i}}$ ${\ displaystyle f \ colon Y \ to X}$ ${\ displaystyle p_ {i} \ circ f = f_ {i}}$ ${\ displaystyle i \ in I}$

Examples

If there are two metric spaces , then the product topology is obtained with the product metric ${\ displaystyle (X_ {1}, d_ {1}), (X_ {2}, d_ {2})}$ ${\ displaystyle X_ {1} \ times X_ {2}}$

{\ displaystyle d ((p_ {1}, p_ {2}), (q_ {1}, q_ {2}))): = {\ sqrt {d_ {1} (p_ {1}, q_ {1}) ^ {2} + d_ {2} (p_ {2}, q_ {2}) ^ {2}}}.}

The product topology on the -fold Cartesian product of the real numbers is the usual Euclidean topology. ${\ displaystyle n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

The product topology on a function space is the topology of point-wise convergence . ${\ displaystyle \ mathbb {R} ^ {M}}$

The Cantor set is homeomorphic to the product space of countably many copies of the discrete space {0, 1}.

The space of the irrational numbers is homeomorphic to the product countable of many copies of the natural numbers with the discrete topology.

The ring of the whole 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 . ${\ displaystyle \ mathbb {Z} _ {p}}$ ${\ displaystyle \ mathbb {Z} / p ^ {n} \ mathbb {Z}}$ ${\ displaystyle \ mathbb {Z} _ {p}}$

properties

The product topology is also called the topology of point-wise convergence because of the following property: A sequence in converges if and only if all projections converge on it. In particular, for the space of all functions from to, the convergence in the product topology is synonymous with the point-wise convergence. ${\ displaystyle X = \ textstyle \ prod _ {i \ in I} X_ {i}}$ ${\ displaystyle X_ {i}}$ ${\ displaystyle \ mathbb {R} ^ {I}}$ ${\ displaystyle I}$ ${\ displaystyle \ mathbb {R}}$

To check whether a given function is continuous, one can use the following criterion: is continuous if and only if all are continuous. Checking whether a function is continuous is usually more difficult; one then tries somehow to exploit the continuity of the . ${\ displaystyle f \ colon Y \ to X}$ ${\ displaystyle f}$ ${\ displaystyle p_ {i} \ circ f}$ ${\ displaystyle g \ colon X \ to Z}$ ${\ displaystyle p_ {i}}$

An important sentence about the product topology is Tichonow's theorem : Every product of compact spaces is compact. This is easy to show for finite products, but the statement is surprisingly also true for infinite products, for the proof of which one then needs the axiom of choice .

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

Others

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

literature

Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .