[Math] Understanding the homotopy extension property

Question

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?

Answer 1

Let's look at Hatcher's definition:

...a homotopy, which is simply any family of maps $f_t : X \to Y , t \in I$ , such that the associated map $F : X × I \to Y$ given by $F(x,t) = f_t(x)$ is continuous. One says that two maps $f_0,f_1: X \to Y$ are homotopic if there exists a homotopy connecting them, and one writes $f_0 \sim f1$

Thus we Hatcher says that for a map $f_0:X \to Y$ with a subspace $A \subset X$ we are given are homotopy $f_t:A \to Y$ of $f_0|A$ this simply means that there is some other map $f_1:A \to Y$ that is homotopic to $f_0|A$. The homotopy extension property then tells us when this homtopy extends to the whole of $X$ (not just the subspace $A \subset X$)

There are plenty of examples. For example Proposition 0.16 gives a useful one - a CW-pair $(X,A)$ has the homotopy extension property.

This will be especially useful when you learn about cofibrations.