Distribution of the y-coordinate of a 2D Poisson Process

Consider a Poisson process with rate 1 over $$R_+ \times R$$. I found this term in a scientific article, I assume it means :

-The number of points in a subset $$A$$ of $$R_+ \times R$$ has the Poisson distribution with paramater the area of $$A$$, and this random variable (number of points in $$A$$) is independant of the number of points in $$B$$ if $$B$$ is disjoint set from $$A$$.

Is this the right definition ?

My main question : it seems that with probability 1, no two points can have the same x-coordinate. Therefore, we can order the points by their x-coordinate : $$(X_1,Y_1),(X_2,Y_2)$$ etc what is the distribution of say the y-coordinate of the first point (the point with smallest x-coordinate)?

I would also want to check if my assumptions are right : the y-coordinate are independant variables, and the x-coordinate form a Poisson Process over $$R$$.

Note : I have heard about the ****2d**** Poisson process over $$R_+ \times [0,M]$$ where $$M$$ is some number, which can be constructed as a marked Poisson process, so the answer to my questions would be yes, but the problem is that here I have $$R_+ \times R$$ instead of $$R_+ \times [0,M]$$ so I can't see my process as a marked Poisson process

• Why should there even exist a point with smallest $x$ coordinate? I believe there is zero chance of this happening, because when it does happen and that coordinate is $x_0\gt 0,$ for all $A\ge 0$ every rectangle $(0, x_0)\times (-A,A)$ has no points and the chance of that is $\exp(-2Ax_0)\to 0$ as $A\to\infty.$ – whuber Apr 9 at 19:22