= External product in topological K-theory
{title2=$\boxtimes:K^0(X)\times K^0(Y)\to K^0(X\times Y)$}
For compact Hausdorff <topological spaces> $X,Y$ and complex <vector bundles> $E\to X$, $F\to Y$, the external product in <Topological K-theory> is
$$
[E]\boxtimes[F]=[p_X^*E\otimes p_Y^*F]\in K^0(X\times Y),
$$
where $p_X,p_Y$ are the two projections. Pullback places both bundles over the same product space, so their fibrewise <tensor product> is defined. The construction respects direct sums and therefore extends bilinearly to differences of bundle classes in the <Grothendieck group>.
For example, external multiplication by the trivial line bundle on a point just pulls the other class back along $X\times\{*\}\to X$. When $X=Y$, pulling the external product back along the diagonal gives the internal ring product on $K^0(X)$. The external product combines information from separate spaces; its existence alone does not assert a Künneth isomorphism for every pair of spaces.
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