My question is,
Is it possible to define geometrical concepts (say, of Euclidean Geometry) like 'point', 'striaght line' in purely set theoretic terms?
So far, I could think of the following definitions,
Let $S$ be a non-empty set.
An element of $S$ is said to be a point.
$S$ will be said to be a plane if there exists two sets $X$ and $Y$ such that, $$S=X\times Y$$
$S$ will be said to be a space if there exists three sets $X$, $Y$ and $Z$ such that, $$S=X\times Y\times Z$$
But the problem is that I couldn't define a straight line in this scheme. This made me wonder if there is any foundation of geometry which is based only on set theory?
If so, then can some related literature be mentioned?