Let $X= X_1 \cup X_2$ a reducible scheme where $X_i$ are it's irreducible components and $Y \subset X$ is an irreducible closed subscheme of $X$. I'm looking for a not too artficial example for $X$ and $Y$ with following properties:
For every open subscheme $U \subset X$ with $U \cap Y \neq \emptyset$ (equivalently: $U \cap Y$ is dense in $Y$, because $Y$ irreducible) the both intersections $U \cap X_1, U \cap X_2$ aren't empty.
in other words every open $U$ that intersects $Y$ non trivially, is dense in both irreducible components $X_i$ simultaneously. this is also equivalent to the property that every open $U$ that intersects $Y$ non trivially is not irreducible.