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

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.

• I'm pretty sure that "typical" as used here is a non-precise descriptive natural language term, and it is not intended to mean all but a measure zero set or all but a first category set (which "typical" is often used in a technical sense for) or all but some other precisely defined small set of exceptions. Basically, it just means that except for highly specialized (or very contrived) situations/cases. Sep 29 at 11:16
• General position Sep 29 at 11:59

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.