let $X$ and $Y$ be topological spaces. Also suppose $A\subseteq Y$ and $f:A\rightarrow X$ is continous. If $q:X+Y \rightarrow X+_f Y $ is the natural projection from the disjoint union of $X$ and $Y$ onto the attachment space.
In Stephen Willard's General Topology, the following is claimed:
The restriction of $q$ to $Y$ is a homeomorphism.
However, this is clearly not true since the restriction is not injective. For example suppose $a,b$ are distinct elements of $A$ such that $f(a) = f(b)$. Then $q$ sends $a$ and $b$ to the same element in the attachment space since we identify $x$ with $a$ and $b$ and so $q(a) = q(b)$.
Am I correct or am I misunderstanding the attachment space construction?
Here is the exact extract from the book,
