Since a general algebraic space does not satisfy this requirement, it allows a single connected component of U to cover X with many "sheets".

With these definitions, the algebraic spaces form a category.

Non-singular algebraic spaces of dimension two (smooth surfaces) are schemes.

Group objects in the category of algebraic spaces over a field are schemes.

Not every non-singular 3-dimensional algebraic space is a scheme.

Thus algebraic spaces are in a sense "close" to affine schemes.

All algebraic spaces are of finite type over a locally Noetherian base.

There are further generalizations called algebraic spaces and stacks.

The algebraic dual space is defined for all vector spaces.

The converse is true in the setting of algebraic spaces.