Topological algebra
In mathematics, a topological algebra is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Definition
A topological algebra over a topological field is a topological vector space together with a bilinear multiplicationthat turns into an algebra over and is continuous in some definite sense. Usually the continuity of the multiplication is expressed by one of the following requirements:joint continuity: for each neighbourhood of zero there are neighbourhoods of zero and such that In the first case is called a "topological algebra with jointly continuous multiplication", and in the last, "with separately continuous multiplication".
A unital associative topological algebra is called a topological ring.