Remark 10.1.3. Given a metric on \(E\), one obtains a homeomorphic description of \(\Th (E)\) as the quotient \(D(E)/S(E)\), where \(D(E)\) and \(S(E)\) denote the unit disk and sphere bundles with respect to the metric. The homeomorphism is given by sending a vector \(v\) to \(v/(1+\|v\|)\) in each fiber. This is often taken as the definition, but has the disadvantage of requiring an auxiliary choice of metric.
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