In topology, the cartesian product of topological spaces can be given several different topologies. The canonical one is the product topology, because it fits rather nicely with the categorical notion of a product. Another possibility is the box topology. The box topology has a somewhat more obvious definition than the product topology, but it satisfies fewer desirable properties.

PropertyValue
dbpprop:abstract
  • In topology, the cartesian product of topological spaces can be given several different topologies. The canonical one is the product topology, because it fits rather nicely with the categorical notion of a product. Another possibility is the box topology. The box topology has a somewhat more obvious definition than the product topology, but it satisfies fewer desirable properties. In general, the box topology is finer than the product topology, although the two agree in the case of direct products (or when all but finitely many of the factors are trivial). Given X such that <math>X := \prod_{i \in I} X_i</math>, or the (possibly infinite) Cartesian product of the topological spaces Xi, indexed by <math>i \in I</math>, the box topology on X is generated by B = { Π Ui | Ui open in Xi}. The name box comes from the case of R, the basis sets look like boxes or unions thereof. It is easily verified that B is actually a basis for the topology. The basis sets in the product topology have almost the same definition as the above, except with the qualification that all but finitely many Ui are equal to the whole space Xi. The product topology satisfies a very desirable property for maps fi : Y → Xi into the component spaces: the product map f: Y → X defined by the component functions fi is continuous if and only if all the fi are continuous. This does not always hold in the box topology, because it is in general a much finer topology, so therefore mapping into the range space makes it much harder for functions to be continuous. This actually makes the box topology very useful for providing counterexamples — many qualities such as compactness, connectedness, metrizability, etc. , if possessed by the factor spaces, are not in general preserved in the product with this topology.
dbpprop:hasPhotoCollection
dbpprop:id
  • 3095 (xsd:integer)
dbpprop:title
  • Box topology
dbpprop:wikiPageUsesTemplate
rdf:type
rdfs:comment
  • In topology, the cartesian product of topological spaces can be given several different topologies. The canonical one is the product topology, because it fits rather nicely with the categorical notion of a product. Another possibility is the box topology. The box topology has a somewhat more obvious definition than the product topology, but it satisfies fewer desirable properties.
rdfs:label
  • Box topology
owl:sameAs
skos:subject
foaf:page
is dbpprop:redirect of
is owl:sameAs of