# Question in Milnor&Stacheff - Characteristic Classes, Construction of Chern Classes

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.

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