# Question on the definition of homotopy relative to a subspace

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 $$t \in [0,1]$$?

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.