I am studying about affine planes An affine plane can be defined as It is an ordered pair ($\mathcal{P}$ ,$\mathcal{L}$), P is non-empty set of points and and L is non-empty collection of the subsets of $\mathcal{P}$ called lines satisfying the following properties

1. Given any two points, there is a unique line joining any two points.

2. Given a point P and a line L not containing P, there is a unique line that contains P and does not intersect L.

3. There are four points, no three of which are collinear. (This rule is just to eliminate the silly case where all of your points are on the same line.) Now what is the problem and what I am thinking about affine plane is the example of Rational affine plane where $$\mathcal{P}=\{(x,y)/x,y\in Q\}$$ and $$\mathcal{L}=\{(x,y)/ ax+by=c\}$$ But in this case the each line in rational affine plane seems to dots. means line is discontineous. Similalry in case of the finite affine planes I am thinking. Can anyone help to remove my this confusion. Thanks in advance
• Do you consider $a,b,c$ real or rational? – user376343 Sep 12 '18 at 11:58
• @Maam Yes $a$, $b$, $c$ are in Q with the condition that $a^2 + b^2 \neq 0.$ – Noor Aslam Sep 12 '18 at 12:00
• don't feel disturbed by "discontinuity" if $\mathcal{P}=\mathbb{Q}×\mathbb{Q}.$ In your new space irrational do not exist, thus there are no holes on lines. As for axioms (1) and (2), they are fulfilled. – user376343 Sep 12 '18 at 12:05
• @Maam I did not get the point please can you explain it further – Noor Aslam Sep 12 '18 at 12:09
• By the way, to get the "vertical bar" in your set descriptions, try using "\mid", like this: \{(x, y) \mid x < y \}, which produces $\{(x, y) \mid x < y \}$, which has nicer spacing and avoids the confusing diagonal slash, which looks like a division symbol. – John Hughes Sep 12 '18 at 13:06