While reading on Wikipedia about non-Euclidean geometry, I tried to think of very simple geometries that invalidate Euclid's first postulate. This lead me to the following question. I do not have any background in geometry beyond high school math. I am asking out of curiosity.
Suppose that $P_a$ and $P_b$ are two disjoint Euclidean planes, and that we do not make any further assumptions about their relation (in particular, I do not assume that these two planes are embedded in a three dimensional Euclidean space). By disjoint I mean that there is no point which is both on $P_a$ and $P_b$.
Both $P_a$ and $P_b$ satisfy all the axioms of Euclidean geometry when considered in isolation. For each, we have the usual definitions of points, lines, angles etc. I was wondering if it make any sense to treat the union of $P_a$ and $P_b$ as a non-Euclidean geometry. In such a union, I assume that a point (line) is in the union iff it is a point (line) in $P_a$ or a point (line) in $P_b$.
Now consider the postulates of Euclidean geometry:
If $p_1$ is a point on $P_a$ and $p_2$ is a point on $P_b$, then there is no line going through $p_1$ and $p_2$. This invalidates Euclid's first axiom ("To draw a straight line from any point to any point", Wikipedia)
Euclid's second and fourth axioms remain valid, as far as I can see.
Playfair's axiom (In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. Wikipedia) is false, if we take the union of $P_a$ and $P_b$ to be a plane. For if you take a line and a point not on it from $P_a$, every line in $P_b$ is parallel to it (where parallel means no intersection at any point). Therefore, there are infinitely many lines parallel to the line on $P_a$. On the other hand, if the union is not a plane, then Playfair's axiom is valid.
My Question:
Does such union geometry make any sense? If no, are there any inconsistencies or abuse of terminology that I am missing? If yes, what is the name of such non-Euclidean geometry and what is its relation to other non-Euclidean geometries?
I think I am confused because it is unclear to me what is the "right way" to generalize concepts from Euclidean geometry, such as the concepts of a plane and parallel lines.
We can ask similar questions about more complicated unions, such as the union of two three dimensional Euclidean spaces, or the union of spherical geometry with a Euclidean plane.