topological product

topological product

[¦täp·ə¦läj·ə·kəl ′präd·əkt]
(mathematics)
The topological space obtained from taking the cartesian product of topological spaces.
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To have more examples, consider the topological product algebra of [E.sub.1] or E with any Banach algebra, commutative or not, unital or not.
Then for an arbitrary base {[e.sub.1], *** , [e.sub.n]}, the strong topology on ([R.sup.n], v, [tau], [[tau].sup.*]) is the topological product of the strong topologies on ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], v, [tau], [[tau].sup.*]), 1 [less than or equal to] i [less than or equal to] n.
If [[alpha].sub.p] = 0 for every [theta] = p [member of] [R.sup.n], then the strong topology on ([R.sup.n], v, [tau], [[tau].sup.*]) is the topological product of strong topologies on all subspaces ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], v, [tau], [[tau].sup.*])(1 [less than or equal to] i [less than or equal to] n), where {[e.sub.1], [e.sub.2], ..., [e.sub.n]} is an arbitrary base of the vector space [R.sup.n].

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