I am just wondering, given the definition of continuous maps as follows,
A functionn $f:X \to Y$ is continuous if for every open subset $U $ of $Y$ the preimage $f^{-1}U$ is open in $X$.
I guess mathematically, this doesn't necessarily mean that "an open subset of $X$ is mapped to an open subset in $Y$"?
It's only that the open subset of $Y$ must originate from an open subset in $X$, but not necessarily that every open $V$ of $X$ will be mapped to some open $U$ of $Y$.
Is this understanding correct?
Best Answer
Yes, that is correct.
A function that maps open sets to open sets is called an open map, i.e a function $f : X \rightarrow Y$ is open if for any open set $U$ in $X$, the image $f(U)$ is open in $Y$.
Open maps are not necessarily continuous.
Then there is the concept of closed maps which maps closed sets to closed sets. A map may be open, closed, both, or neither and continuity is independent of openness and closedness.
A continuous function may have one, both, or neither property.