Pullback bundle

In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle ${\displaystyle \pi :E\rightarrow B}$ and a continuous map ${\displaystyle f:B'\rightarrow B}$ one can define a "pullback" of ${\displaystyle E}$ by ${\displaystyle f}$ as a bundle ${\displaystyle f^{*}E}$ over ${\displaystyle B'}$. The fiber of ${\displaystyle f^{*}E}$ over a point ${\displaystyle b'}$ in ${\displaystyle B'}$ is just the fiber of ${\displaystyle E}$ over ${\displaystyle f(b')}$. Thus ${\displaystyle f^{*}E}$ is the disjoint union of all these fibers equipped with a suitable topology.