one-sided projective plane vs two-sided projective plane

Question

This Wikipedia page contains the definition of a projective plane. But when reading a paper, I saw two kinds of projective planes: one-sided and two-sided projective planes. They did not write any definition, and I tried to google but did not find any textbooks that talk about those. Can anyone explain the difference between the "projective plane", "one-sided projective plane" and "two-sided projective plane"? What I know is that a cylinder is a two-sided surface, the Mobius band is a one-sided surface.

Answer 1

Most likely, you see this notion in the context of 3-dimensional topology. In order to simplify things, I will work in the category of smooth manifolds.

Definition. Suppose that $M$ is a (smooth) 3-dimensional manifold. A properly embedded connected surface $S\subset M$ is called 2-sided if it locally separates $M$. Equivalently, there exists a smooth vector field along $S$ which is everywhere normal to $S$. Otherwise, a connected surface is called 1-sided.

A projective plane can be either 2-sided or 1-sided in a 3-dimensional manifold. An example of a 1-sided projective plane is $RP^2$ embedded in $RP^3$. An example of a 2-sided projective plane in a 3-manifold is $RP^2\times \{0\}\subset RP^2\times (-1,1)$.

Actually, even a cylinder can be 1-sided in a certain 3-manifold, called solid Klein bottle.

See also page 10 in these notes on 3-dimensional topology by Alan Hatcher.

See also my answer here for a more detailed explanation of the notion of a side of a surface.