Proposition 2

Let B and $B'$ be topological spaces, let $(E, p)$ be a B-space and let $(E', p')$ be a $B'$-space. Suppose that E and $E'$ are locally trivial fibre spaces ( resp. coverings). The space $E\times E'$, endowed with the mapping $p\times p': E\times E'\rightarrow B\times B'$, is then a locally trivial fibre space ( resp. a covering) with base $B\times B'$. If E and $E'$ are trivializable, so is $E\times E'$.

Suppose first that the B-space E and the $B'$-space $E'$ admit trivializations $g: E\rightarrow B\times F,g': E'\rightarrow B'\times F'$; the composite of the homeomorphism $g\times g'$ of $E\times E'$ onto $(B\times F)\times (B'\times F')$ and of the canonical homeomorphism of $(B\times F)\times (B'\times F')$ onto $(B\times B')\times (F\times F')$ is a trivialization of the $B\times B'$-space $E\times E'$, of fibre-type $F\times F'$. If F and $F'$ are discrete spaces, the space $F\times F'$ is also discrete.

In the general case, for every point $(a, a')$ of $B\times B'$, there exists a neighbourhood U of $a$ in B and a neighbourhood $U'$ of $a'$ in $B'$ such that the U-space $E_U$ and the $U'$-space $E'_{U'}$ are trivializable. It follows from the foregoing that the $B\times B'$-space $E\times E'$ is trivializable over $U\times U'$. Hence it is a locally trivial fibre space with base $B\times B'$. Its fibres are discrete if E and $E'$ are coverings, whence the proposition.

It should be observed that, if $(p_i: E_i\rightarrow B_i)_{i\in I}$ is a family of locally trivial fibre spaces, or even of coverings, the product space $\prod_{i\in I}E_i$ endowed with the mapping $(p_i)_{i\in I}$ is not necessarily a locally trivial fibre space with base $\prod_{i\in I}B_i($I, p. 145, Exercise 3).