What are the subsets of a line define each?

Answer 1

The various subsets are:

• The empty set: nothing.
• A point: a position on the line with 0 dimensions.
• A line segment: a 1-dimensional finite subset such that, if x and y are any two points in the subset then so is mx + (1-m)y from any m in [0,1]. That is, all points between x and y are also in the set.
• A ray: a 1-dimensional subset such that one of the points x and y is an infinite distance away.
• Collections of a finite or infinite number of points, line segments and rays.
• The line itself.

For lines and rays, the end points may or may not be part of the subset.