Is every homeomorphism a quotient map? Let $X$ and $Y$ be topological spaces. Let $f: X \rightarrow Y$ be a bijections such that $f(U)$ is open in $Y$ if and only if $U$ is open in $X$. Then $f$ is a homeomorphism and an open map. In particular, it is a surjective open map and, therefore, it is a quotient map, right? 
Now let $X$ be a topological space and let $Y$ be the set of point in the topological space $Y$ above. Munkre's states that if $f:X \rightarrow Y$ is a surjective map then there is exactly one topology on $Y$ relative to which $f$ is a quotient map and it's the quotient topology defined by: $U$ is open in $Y$ if $f^{-1}(U)$ is open in $X$.
Therefore, I deduce that if $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a homeomorphism, then $f$ is a quotient map and $Y$ has the quotient topology. This seems fishy. Is it correct or incorrect? 

Everything you said is true, but it's kind of an abuse of the definitions.
We like to think of a a quotient map as somehow gluing things together, and a homeomorphism corresponds to the trivial case where we glue together nothing. Moreover, in a homeomorphism, we usually think of $Y$ as being a space in its own right, whereas the quotient topology is more like a structure that the function imposes on the space. (Anthropomorphizing, we might say that the function "wants" to be continuous and so it "demands" that $Y$ have a particular topology).
