I was reading this translation of Analysis Situs and on the first page (page 15 in the pdf), the following phrase comes up.
Let $x_1,x_2\dots,x_{n+1}$ be the coordinates of a point on the surface. These $n+ 1$ quantities are connected by the equation of the surface.
Is there an implicit assumption that a 'surface' is a variety? Or does the phrase have some other meaning