In this definition (early on):
He never specifies what $f_t | A = 1$ is supposed to mean in his preamble. What is $|$ being used for in this example?
In "Standard Notations" on p. xii Hatcher writes
: the identity function from a set to itself.
The vertical "$\mid$" is the standard notation for the restriction of a function. That is, $f_t\mid A$ denotes the restriction of $f_t : X \to X$ to the subspace $A \subset X$.
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In "Standard Notations" on p. xii Hatcher writes
The vertical "$\mid$" is the standard notation for the restriction of a function. That is, $f_t\mid A$ denotes the restriction of $f_t : X \to X$ to the subspace $A \subset X$.