Thom space as a one-point compactification

Remark 10.1.4. If \(X\) is compact Hausdorff, the inclusion of \(E\) induces a homeomorphism \(E^+\cong \Th (E)\). Indeed, a neighborhood of the collapsed section in \(\Th (E)\) has compact complement because \(S^E\) is compact Hausdorff, so the quotient topology at the basepoint is the one-point compactification topology.