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The product space , together with the canonical projections, can be characterized by the following universal property: if is a topological space, and for every ,: is a continuous map, then there exists precisely one continuous map : such that for each the following diagram commutes:
Another possibility is the product topology, where a base is also given by the Cartesian products of open sets in the component spaces, but only finitely many of which can be unequal to the entire component space. While the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirable properties.
Given an H-space with multiplication :, the Pontryagin product on homology is defined by the following composition of maps (;) (;) (;) (;)where the first map is the cross product defined above and the second map is given by the multiplication of the H-space followed by application of the homology functor to obtain a homomorphism on the level of homology.
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.
In differential topology, any fiber bundle includes a projection map as part of its definition. Locally at least this map looks like a projection map in the sense of the product topology and is therefore open and surjective. [citation needed] In topology, a retraction is a continuous map r: X → X which restricts to the identity map on its ...
The homotopy groups of a product space are naturally the product of the homotopy groups of the components, ((,) (,)) ((,)) ((,)), with the isomorphism given by projection onto the two factors, fundamentally because maps into a product space are exactly products of maps into the components – this is a functorial statement.
More abstractly, the outer product is the bilinear map (,) sending a vector and a covector to a rank 1 linear transformation (simple tensor of type (1, 1)), while the inner product is the bilinear evaluation map given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.
The definition works without any changes if instead of vector spaces over a field F, ... carries an inner product, then the inner product is a bilinear map ...