I've just started working through a topology textbook, and I'm sure I'm being silly, but I can't for the life of me make any headway on the following question:
Let $(M,\rho)$ be a pseudometric space, $(M^*,\rho^*)$ its metric identification, $h(x)$ the canonical projection map from $M\to M^*$.
Then a set $A$ in $M$ is closed (open) iff $h(A)$ is closed (open) in $M^*$.
I have $A$ in $M$ open $\implies$ $h(A)$ open in $M^*$.
The thing that's tripping me up is considering say:
$\{1,2\}$ with the trivial pseudometric. The metric identification is $\{[1]\}$, so $\{[1]\}$ is open in the metric identification, but $\{1\}$ is not open in the pseudometric space, since every disk around $1$ contains $2$ as well.
Thanks I'm sure I'm probably doing something stupid, but after googling it I couldn't find an explanation so I came here.
The characterization of the topology on $M$ that you have written (which, I see, is the characterization on the wikipedia page for pseudometrics) has an error.
It should be: "a set $A$ in $M$ is closed (open) iff $A$ is saturated and $h(A)$ is closed (open) in $M^*$".