Why do we say $n$ distinct points?

Question

" Let's say we have $n$ distinct points... " , you see this every time you open a geometry textbook. Why not just $n$ points ? If the points are not distinct, they are not exactly $n$ points, are they ? We don't write a set as $ \{x_1,x_2,x_3\}$ where $x_1 =x_2$, or do we ? Can you give me an example of a case in geometry, where an ambiguity might arise if one does not do so ?

Answer 1

This is done to clearly state that the points are distinct.

Note that it is not written that we have a set of $n$ points and so without writing that the points are distinct it may be interpreted that the statement being made is of the form $$ \forall p_{1},p_{2},...,p_{n}:\text{[statement]} $$

where the $p_{i}$ can be any points, without the restriction that they are different.

Answer 2

Because, in the following discussion, the actual number of points occurs and needs to be referred to.

For example, it might say "Let the points be $(P_i)_{i=1}^n$."

Answer 3

When it comes to points in relation to curves (as well as to eigenvalues and to zeros of polynomials) neither set nor sequence conveys the right idea. The concept needed is that of bag: a set with an associated multiplicity for each element. For example, consider the points where two algebraic curves of degree $n_1$ and $n_2$ intersect. Typically there are $n_1n_2$ points of intersection (including complex values). But, in special cases, some of these points coincide, where the curves are tangent or have multiple crossings; the number of coincident points indicates the order of tangency or multiple intersection. If we only consider the points as a set, we have fewer then $n_1n_2$ of them, and information about where the tangencies or multiple crossings lie is lost. If there are $n$ points, and no tangency or multiple intersection is involved, we might say "$n$ points of single multiplicity". But this is making rather a meal of it; saying "$n$ distinct points" sounds lighter.