Thom space of an external direct sum

Example 10.1.6. Let \(E \to X\) and \(E' \to X'\) be vector bundles. Then their product \(E \times E' \to X \times X'\) is again a vector bundle (called their external direct sum) and there is an isomorphism of pointed spaces \[ \Th (E \times E') \cong \Th (E) \wedge \Th (E'). \] To see this, equip \(E \times E'\) with the metric \(\|(v,w)\| = \max (\|v\|,\|w\|)\). Then the disk bundle \(D(E \times E')\) is homeomorphic to the product \(D(E) \times D(E')\), and under this homeomorphism the sphere bundle \(S(E \times E')\) corresponds to the subspace \[ D(E)\times S(E') \cup S(E) \times D(E') \] of the product. Passing to quotient spaces then proves the claim.