Definition 1

A topological B-space (or simply a B-space) is a topological space X, endowed with a continuous mapping $p$ of X into B. The mapping $p$ is called the projection of the B-space X.

Let X, $X'$ be B-spaces and $p,p'$ their respective projections. A B-morphism of X into $X'$ is a continuous mapping $f$ of X into $X'$ such that $p'\circ f=p$.

It is sometimes convenient to denote by $(X, p)$ the B-space obtained by endowing the topological space X with the continuous mapping $p$.

The composite of two B-morphisms is a B-morphism. The isomorphisms of B-spaces, also called B-isomorphisms, are the B-morphisms which are homeomorphisms.

Thus, if one calls a B-space structure on a set X the datum of a topology on X and of a continuous mapping $p: X\rightarrow B$, one can take the B-morphisms as morphisms of the B-space structure (E, IV, p. 11).

© N. Bourbaki and Springer-Verlag Berlin Heidelberg 2 016

N. Bourbaki, Algebraic Topology, DOI 10.1007/978-3-662-49361-8_1 1

Let X, $X'$ be B-spaces. We denote by $\mathscr{C}_B(X; X')$ the set of B-morphisms of X into $X'$, by Isom$_B(X; X')$ the set of B-isomorphisms of X into $X'$, and by Aut$_B(X)$ the set of B-automorphisms of X, that is to say of B-isomorphisms of X into X.

Let X be a B-space and $p$ its projection. If B is endowed with the B-space structure whose projection is Id$_B$, the set $\mathscr{C}_B(B; X)$ is the set of continuous sections (E, II, p. 18, Definition 11) of $p$.

Let X be a B-space, $p$ its projection and $b$ a point of B. The subspace $\overset{-1}{p}(b)$ of X is called the fibre of X at $b$ (or the fibre of $p$ at $b$) and denoted by $X_b$. In order that a continuous mapping $f$ of X into a B-space $X'$ be a B-morphism, it is necessary and sufficient that $f(X_b)$ be contained in $X'_b$ for every $b\in B$.

Let X be a B-space and $p$ its projection. Let $f$ be a continuous mapping of a topological space $B'$ into B. A continuous mapping $g: B'\rightarrow X$ such that $p\circ g=f$ is called a continuous lifting of $f$ to X. In other words, if $B'$ is endowed with the B-space structure of projection $f$, the continuous liftings of $f$ to X are the B-morphisms of $B'$ into X.