Let $X = (\mathbb R, d)$ be the usual real line and $Y =(\mathbb R, d')$ be the set R with discrete metric.
Show that identity map from $X$ to $Y$ is not continuous but open as well as closed.
On the other hand, the identity map from $Y$ to $X$ is continuous which is neither open nor closed.
My attempt :
we know that for continuity we want to show that inverse image of the open set is open.
Any singleton set $\{x\}$ is open in discrete metric space and hence its inverse image under identity map is also $\{x\}$, which is not an open set in usual metric.
Hence, identity map is not continuous.
Again if identity map have domain with discrete metric then it is always continuous.
Am I correct??
Now for open map (closed), we want to show that image of open (closed) set is open (closed).
How to use this definition to prove open and closed.
Please help. Thank you.
Best Answer
What you did is correct.
Now, you have to keep in mind that, with respect to the discrete metric every set is open and every set is closed. In fact, given a set $S$, $S=\bigcup_{x\in S}\{x\}$ and, since each singleton is open, $S$ is open. And since every set is open, every set is closed too. Therefore, every map from an arbitrary metric space into a discrete one is both open and closed.
In order to prove the the identity from $Y$ to $X$ is neither open nor closed, you can take, for instance, the set $A=[0,1)$. It is neither open nor closed in $X$ but it is both open and closed in $Y$. Since $\operatorname{id}(A)$, $\operatorname{id}$ is neither open nor closed.