In everyday English, the word "sphere" denotes a 3-dimensional object, including the points inside the surface and its center.
However, I get the sense that in mathematics, the sphere is used predominantly to denote the set of points on the surface of such an object. It is a 2-dimensional surface that does not include interior points. Is this correct?
On
Just wanted to add that mathematicians have generalized the idea of a sphere to any number of dimensions, i.e., one possible way to define a 3-sphere, a higher dimensional analogue of the ordinary sphere is $$ x^2+y^2+z^2+w^2=a^2 $$ which is the set of all points equidistant from the origin in a 4 dimensional space.
One can then ask: what, fundamentally, makes something a sphere? This question turns out to be non-trivial to answer, but topologists now have an answer to that question in every dimension.
This is correct. In standard mathematical usage, "sphere" denotes the surface only. The term for the interior is instead "ball" (more precisely, an "open ball" is just the interior, while a "closed ball" is the interior together with the surface).