Question on the definition of homotopy relative to a subspace

Question

I am beginning to learn algebraic topology from Hatcher's text and I had a question about the definition of homotopy relative to a subspace. The passage in the text goes as follows:

A homotopy $f_{t}: X \rightarrow X$ that gives a deformation retraction of $X$ onto a subspace $A$ has the property that $f_{t}|_{A} = id$ for all $t$. In general, a homotopy $f_{t}: X \rightarrow Y$ whose restriction to a subspace $A \subset X$ is independent of $t$ is called a homotopy relative to $A$.

I do not understand exactly what is meant by "whose restriction to a subspace $A \subset X$ is independent of $t$." From what I gathered reading the wikipedia article, I am guessing this means that $f_{t}(a) = id(a) = a$ for all $a \in A$ and $t \in [0,1]$?

Answer 1

You are mixing the concept of a deformation retraction onto a subspace with the more general notion of homotopy relative to a subspace.

In the latter case, for each $a\in A$, you want the image of $a$ under $f_t$ to be the same for any $t$; but $f_t$ need not be the identity on the subspace. In fact, $f_t$ being the identity does not even make sense in general as $f_t$ is a map between two (possibly different) topological spaces $X$ and $Y$.

In the specific case of a deformation retraction, where $f_t$ is a map from $X\to X$ and restricts to the identity on the subspace $A\subset X$, we have $f_t(a)=id(a)=a$ as you have written.