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The following paragraph is extracted from the book:

We will now give an inductive definition of characteristic classes for a complex $n$-plane bundle $\omega=(\pi: E\to M)$. If is first necessary to construct a canonical $(n-1)$-plane bundle $\omega_0$ over the deleted total space $E_0$. ($E_0$ denotes the set of all nonzero vectors in $E$.) A point in $E_0$ is specified by a fiber $F$ of $\omega$ together with a nonzero vector $v$ in that fiber. First suppose a Hermitian metric has been specified on $\omega$. Then the fiber of $\omega_0$ is by definition, the orthogonal complement of $v$ in the vector space $F$. This is a complex vector space of dimension $n-1$, and these vector spaces clearly can be considered as the fibers of a new vector bundle $\omega_0$ over $E_0$.

Question: I understood how the total space of $\omega_0$ is defined. But how is the topology of the total space defined? There is no mention about it.

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1 Answer 1

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Consider the following mappings:

$\require{AMScd}$ \begin{CD} \pi^*E @>>> E\\ @V \bar\pi V V @VV \pi V\\ E @>>\pi> M \end{CD}

which induces a pullback bundle $\bar \pi : \pi^*E \to E$, where for each $v\in E$, $$\bar\pi^{-1} (v) = \pi^{-1} (\pi(v)).$$ (that is, the fiber is just the fiber $F_x$, where $x = \pi(v)$). $\pi^*E$ is given the topology of the pullback bundle. Since $E_0$ is a subset of $E$, the restriction gives a bundle

$$\tag{1} \bar\pi \big|_{\bar\pi^{-1}(E_0)} : \pi^*E\big|_{\bar\pi^{-1}( E_0)} \to E_0$$

and the bundle $\omega_0$ constructed in the book is a sub-bundle of (1). Plus is has the subspace topology given by (1).

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