# Fibre space

An object $(X,\pi,B)$, where $\pi: X \to B$ is a continuous surjective mapping of a topological space $X$ onto a topological space $B$ (i.e., a fibration). Note that $X$, $B$ and $\pi$ are also called the total space, the base space and the projection of the fibre space, respectively, and ${\pi^{\leftarrow}}[\{ b \}]$ is called the fibre above $b$. A fibre space can be regarded as the union of the fibres ${\pi^{\leftarrow}}[\{ b \}]$, parametrized by the base space $B$ and glued by the topology of $X$. For example, there is the product $\pi: B \times F \to B$, where $\pi$ is the projection onto the first factor; the fibration-base $\pi: B \to B$, where $\pi = \operatorname{id}$ and $X$ is identified with $B$; and the fibre space over a point, where $X$ is identified with a (unique) space $F$.

A section of a fibration (fibre space) is a continuous mapping $s: B \to X$ such that $\pi \circ s = \operatorname{id}_{B}$.

The restriction of a fibration (fibre space) $\pi: X \to B$ to a subset $A \subseteq B$ is the fibration $\pi': X' \to A$, where $X' \stackrel{\text{df}}{=} {\pi^{\leftarrow}}[A]$ and $\pi' \stackrel{\text{df}}{=} \pi|_{X'}$. A generalization of the operation of restriction is the construction of an induced fibre bundle.

A mapping $F: X \to X_{1}$ is called a morphism of a fibre space $\pi: X \to B$ into a fibre space $\pi_{1}: X_{1} \to B_{1}$ if and only if it maps fibres into fibres, i.e., if for each point $b \in B$, there exists a point $b_{1} \in B_{1}$ such that $F[{\pi^{\leftarrow}}[\{ b \}]] \subseteq {\pi^{\leftarrow}}[\{ b_{1} \}]$. Such an $F$ determines a mapping $f: B \to B_{1}$, given by $f(b) \stackrel{\text{df}}{=} (\pi \circ F)[{\pi^{\leftarrow}}[\{ b \}]]$. Note that $F$ is a covering of $f$ and that $\pi_{1} \circ F = f \circ \pi$; the restrictions $F_{b}: {\pi^{\leftarrow}}[\{ b \}] \to {\pi_{1}^{\leftarrow}}[\{ b_{1} \}]$ are mappings of fibres. If $B = B_{1}$ and $f = \operatorname{id}$, then $F$ is called a $B$-morphism. Fibre spaces with their morphisms form a category — one that contains fibre spaces over $B$ with their $B$-morphisms as a subcategory.

Any section of a fibration $\pi: X \to B$ is a fibre-space $B$-morphism $s: B \to X$ from $(B,\operatorname{id},B)$ into $(X,\pi,B)$. If $A \subseteq B$, then the canonical imbedding $i: {\pi^{\leftarrow}}[A] \to B$ is a fibre-space morphism from $\pi|_{A}$ to $\pi$.

When $F$ is a homeomorphism, it is called a fibre-space isomorphism. A fibre space isomorphic to a product is called a trivial fibre space. An isomorphism $\theta: X \to B \times F$ is called a trivialization of $\pi$.

If each fibre ${\pi^{\leftarrow}}[\{ b \}]$ is homeomorphic to a space $F$, then $\pi$ is called a fibration with fibre $F$. For example, in any locally trivial fibre space over a connected base space $B$, all the fibres ${\pi^{\leftarrow}}[\{ b \}]$ are homeomorphic to one another, and one can take $F$ to be any ${\pi^{\leftarrow}}[\{ b_{0} \}]$; this determines homeomorphisms $\phi_{b}: F \to {\pi^{\leftarrow}}[\{ b \}]$.

Both the notations $\pi: X \to B$ and $(X,\pi,B)$ are used to denote a fibration, a fibre space or a fibre bundle.
In the West, a mapping $\pi: X \to B$ would only be called a fibration if it satisfied some suitable condition, for example, the homotopy lifting property for cubes (such a fibration is known as a Serre fibration; see Covering homotopy for the homotopy lifting property ([a3])). A mapping $F: X \to X_{1}$ would be called a morphism (respectively, an isomorphism) only if the induced function $f: B \to B_{1}$ were continuous (respectively, a homeomorphism).