I'm reading Chapter 0 of Hatcher right now, and there's something in the definition of homotopy extension property that I don't understand.
Suppose one is given a map $f_0:X\to Y$, and on a subspace $A\subset X$ one is also given a homotopy $f_t:A\to Y$ of $f_0|_A$ that one would like to extend to a homotopy $f_t:X\to Y$ of the given $f_0$. If the pair $(X,A)$ is such that this extension problem can always be solved, one says that $(X,A)$ has the homotopy extension property.
What does it mean to say that we have a "homotopy… of $f_0|_A$" ? By Hatcher's definition, a homotopy is just a family of maps (with a continuity condition), and that two maps may be considered homotopic…. But what is a homotopy of a single map, exactly?
Also, does anyone have a nice intuitive example of a pair with this property?