[Math] $f$ mapping open sets to open sets

Question

I know that the definition of a continuous mapping between two topologies is defined as:

For $\mathcal{X}$ and $\mathcal{Y}$ and $f$ such that $f:\mathcal{X}\rightarrow \mathcal{Y}$ if $f^{-1}$ maps open sets in $\mathcal{Y}$ to $\mathcal{X}$ then $f$ is continuous.

However I was wondering if instead of $f^{-1}$ mapping open set to open sets what if $f$ maps open sets to open sets?

I know that this does not define continuity but does it define anything that is in anyway useful? (It obviously defines something)

I mean I can see that any homeomorphism satisfies this property so I'm thinking its maybe not that important?

Thanks for any help

Answer 1

As mentioned in the comments, if $X$ and $Y$ are topological spaces and $f : X \to Y$ has the property that $f(U)$ is open for every $U \subseteq X$ open, then $f$ is called an open map.