Definition 11.1 (Homotopies of maps):

(i) Given continuous maps $f, g : X \longrightarrow Y$ between topological spaces we say that $f$ and $g$ are homotopic if there exists a continuous map $F : X\times [0, 1] \longrightarrow Y$ such that

$$F(x, 0) = f(x),\quad F(x, 1) = g(x),$$    for all $$x \in X \eqno(11.1)$$

We shall occasionally use the notation $f \sim g$ to indicate that $f$ and $g$ are homotopic. One can formulate a notion for pairs of spaces:

(ii) Two continuous maps $f, g : (X, A)\longrightarrow (Y, B)$ between pairs of topological spaces are said to be homotopic if there exists $F: (X\times I, A\times I)\longrightarrow (Y, B)$ such that in addition to (11.1) the following condition holds:

$$F(a, t) \in B,$$    for all $$a \in A, t \in [0, 1]. \eqno(11.2)$$

Condition (11.2) is a boundary condition which states that the intermediate functions

$$F_t: x\mapsto F(x, t)$$

all map $A$ into $B$. Note that when $A = \{x_0\}$ and $B = \{y_0\}$, the condition says that all the intermediate maps $F_t$ are base point preserving. We leave it to the reader to prove the following two simple results.