# Generalizing surfaces

In algebra we added arithmetic properties to sets cumulatively until we had built enough structure to arrive at the real numbers. We then went further by considering generalizations of vectors. In geometry, we build up to a surface in three-dimensional space, and then go further by considering generalizations of tangent vectors to a surface.

In this section we will give a preview of the basic ideas, with the implicit assumption that the reader already has some familiarity with objects and relations that will only be defined later.

Euclidean surface Embedding in $${\mathbb{R}^{3}}$$ Relation to a higher space