Definition 1

A pointed set is a set X endowed with one of its elements, called the base point. The set X endowed with the point $x$ is sometimes denoted by $(X, x)$. Let $(X, x)$ and $(Y, y)$ be pointed sets; a pointed mapping of $(X, x)$ into $(Y, y)$ is a mapping $f$ of X into Y such that $f(x) =y$.

The notions of pointed topological space, continuous pointed mapping, pointed covering of a pointed topological space, etc., are defined analogously.

If $(X, x)$ and $(Y, y)$ are pointed topological spaces, the set of continuous pointed mappings of $(X, x)$ into $(Y, y)$ is denoted by $\mathscr{C}((X, x); (Y, y))$.

Instead of saying “let $f$ be a pointed mapping of $(X, x)$ into $(Y, y)$”, one often uses the following phrase: “let $f: (X, x)\rightarrow (Y, y)$ be a pointed mapping”.

Let B be a topological space and let $b$ be a point of B.