I am trying to to understand an example from Munkres book which illustrates that product of two quotient maps may not be quotient map.
I have spent a whole day trying to understand this proof and eventually I have two questions which I cannot answer by myself.
1) Since $X=\mathbb{R}$ with usual topology and $X^*$ its partition with elements $\{\mathbb{Z}_+, \{x\}\}$, i.e. $\mathbb{Z}_+$ and other singletons. I cannot understand two moments here:
1.1. What does mean indentifying the subset $\mathbb{Z}_+$ to a point $b$? Can anyone give a rigorous meaning to this phrase?
1.2. Also how looks like the function $p:X\to X^{*}$?
2) Why it is obvious that the set $U$ is saturated with respect to $(p,i)$? I see that $\mathbb{Z}_+\times \{q\} \subset U$ for each $q\in \mathbb{Q}$. But how it follows that this set $U$ is saturated?
I would be very grateful if you can answer my questions with details.
Best Answer
1) Munkres explains this in the definition on p.139. In the present case the partition $X^*$ has only one non-trivial (which means non-singleton) element. More generally, "identifying a subset $A$ of a space $Y$" to a point yields a space usually denoted as $Y/A$. Its elements are the sets $\{ y \}$ with $y \notin A$ and the set $A$. The quotient map $p : Y \to Y/A$ is given by $p(y) = \{ y \}$ für $y \notin A$ and $p(y) = A$ for $y \in A$.
2) Let us more generally consider an arbitrary surjection $r : Y \to Z$. Then a subset $M \subset Y$ is saturated if and only if for all $y \in M$ we have $r^{-1}r(y) \subset M$.
Let us look at $r = p \times i$. If $(x,q) \in X \times \mathbb Q$, then $r^{-1}r(x,q) = (p \times i)^{-1}(p(x),q) = p^{-1}p(x) \times \{q\}$. You know that $\mathbb{Z}_+\times \{q\} \subset U$ for each $q\in \mathbb{Q}$. Let $(x,q) \in U$. If $x \in \mathbb Z_+$, then $r^{-1}r(x,q) = \mathbb{Z}_+\times \{q\} \subset U$. If $x \notin \mathbb Z_+$, then $r^{-1}r(x,q) = \{x\} \times \{q\} \subset U$.