The Pointed Theory

Abstract

In this chapter we work over a pointed base space B. A fibrewise pointed space over B consists of a space X together with maps

$$ B\xrightarrow{s}X\xrightarrow{p}B $$

such that p ∘ s = 1 _B . In other words, X is a fibrewise space over B with section s. The alternative terminology sectioned fibrewise space is also widely used. Note that the projection is necessarily a quotient map and the section is necessarily an embedding. It is often convenient to regard B as a subspace of X so that the projection retracts X onto B. To simplify the exposition in what follows let us assume, once and for all, that the embedding is closed, as is necessarily the case when X is a Hausdorff space. We regard any subspace of X containing B as a fibrewise pointed space in the obvious way; no other subspaces will be admitted. When s is a cofibration we describe X as cofibrant.