The exercise is number 6 of section 5 Open and closed maps, first chapter. It says:
Let $S$ be the equivalence relation on the rational line $\mathbb Q$ which is obtained by identifying all the points $\mathbb Z$. Show that $S$ is closed, and that if $I$ is the relation of equality on $\mathbb Q$, then the canonical bijection of $(\mathbb Q\times\mathbb Q)/(I\times S)$ onto $\mathbb Q\times (\mathbb Q/S)$ is not a homeomorphism.
The canonical bijection refers to the bijection that appears in the decomposition of a map $f:X\rightarrow Y$ as $\mu:X\rightarrow X/R_f$, $h:X/R_f\rightarrow f(X)$ and $\iota:f(X)\rightarrow Y$. Here $R_f$ denotes the equivalence relation $x\sim y$ if and only if $f(x)=f(y)$.
In this problem, the $f$ should be the product map $id\times\mu_S$, where $\mu_S$ is the quotient map of $\mathbb Q$ into $\mathbb Q/S$. The equivalence relation $I\times S$ can be viewed as the induced by the product map. Moreover, in this case the injection $\iota$ is just the identity map on $\mathbb Q\times(\mathbb Q/S)$. Finally, the canonical bijection $h$ takes an element $[(x,y)]$ of the quotient and give the element $(x,[y])$.
The problem I see here is that for me the product map is closed and hence by a previous theorem this is equivalent to say that $h$ is a homeomorphism. So, can you explain me why the maps $id\times \mu_S$ and $\mu:\mathbb Q\times\mathbb Q \rightarrow \mathbb (\mathbb Q\times\mathbb Q)/(I\times S)$ are not closed? (It is clear that they are not open).
Thanks in advance.
Take a sequence $P_n=(1/n, n)$. Clearly $A=\{P_n\}_{n>0}\subseteq\mathbb{Q}\times\mathbb{Q}$ is a closed subset because it is discrete. But is the image under $\mu$ or $id\times\mu_S$ closed?