My lecturer writes in my topology notes that :
A map $f:(X,\tau)\rightarrow (Y,\tau')$ is a homeomorphism if it is a bijection and both $f,f^{-1}$ are continuous.
However he then goes on to write that :
If $f$ is homeomorphism then $f$ and it's inverse $f^{-1}$ map open sets to open sets and we call them open maps and that we should note that open maps are not necessarily continuous.
He gives the example :
Let $X=\{a,b\}$, consider $id_x:(X,\text{trivial topology})\rightarrow(X,\text{discrete topology})$
This is a bijection.
It's inverse is continuous
But the map itself is not continuous
$id_x(\{a\})=\{a\}$ is not open in the trivial topology.
I'm having trouble seeing how this is consistent with the definition though, could anyone please help me understand ?
Every set is open in discrete topology, so $\{a\}$ is open in $X$ with discrete topology. The inverse image of this set under the identity map is $\{a\}$ is itself. But the only open sets in trivial topology are the empty set and $X$. Hence (assuming $X$ is not a singleton) the inverse image is not open so the identity map is not continuous.
Continuity of the inverse map follows from the fact that every set is open in discrete topology.