What does it mean to say that three "typical" points in the real plane are non-collinear?

Question

I have heard several math books claim that three "typical" points in $\mathbb{R}^2$ are non-collinear. What does this mean? Does it have something to do with measure theory? Of course, one could ask the same thing about the claim that four typical points in $\mathbb{R}^3$ are non-coplanar, and even higher-dimensional analogues. I would be very interested in the precise and rigorous meaning of that claim, as well as the proof of it.

Answer 1

Let $C$ be the set of collinear triples in $\mathbb{R}^3$. When we give $\mathbb{R}^3$ its usual topology and measure, the set $C$ is "small" in both senses of measure and category: the Lebesgue measure of $C$ is zero, and $C$ is meager (indeed nowhere dense). These are each good exercises.

In general, we can talk about typicality using either measure or category. The issue is that for complicated properties we get disagreements between the two notions - e.g. "measure-typically" real numbers are normal, but "category-typically" reals aren't - but for fairly simple properties the two notions agree.