Functions continuous for a topology within a compactification I wonder if the concept of the title, explained in more details thereafter, has a name.
For a topological space $X$, write $X'$ for its compactification (the kind of which is not relevant here).  Consider a continuous function $f\colon X' \to Y'$; is there a name for the following property?

For any open $C$ of $Y$, $f^{-1}(C) \cap X$ is an open of $X$

 A: First, a couple of observations:
(0) If $X$ is discrete then your property holds.
(1) If $Y$ is locally compact, then your property holds. For then $Y$ is open in $Y'$. So if $C$ is open in $Y$, then $C$ is also open in $Y'$, so $f^{-1}[C]$ is open in $X'$, and finally $f^{-1}[C]\cap X$ is open in $X$.
(2) If $X=Y$ and $f\restriction X=\text{id}_X$, then the property also holds.  Because then you must have $f[X'\setminus X]\subseteq Y'\setminus Y$ (prove as an exercise), and so $C=f^{-1}[C]\cap X$.
Now I would like to redo your property a bit. Say $P(X,Y)$ holds if for every two compactifications $X'$ and $Y'$ of $X$ and $Y$ respectively, and every continuous surjection $f:X'\to Y'$, we have that $f^{-1}[C]\cap X$ is open in $X$ whenever $C$ is open in $Y$.
My question then would be: 

Is there a non-discrete space $X$ and a non-locally compact space $Y$ such that $P(X,Y)$ holds nontrivially (meaning $X$ and $Y$ are Tychonoff spaces and some $f$'s actually exist)?

By the way, I don't know if $P$ has been studied before. It may be interesting.
