Examples of topological Immersion which are not embedding

It is said that topological immersion is locally 1-1 while embedding is globally 1-1, so we can say topological immersion is locally ’embedding’? (this also means except for a few points the map is 1-1, and for these points we can find a neighborhood on which the map becomes embedding?)

And embedding basically means we map a manifold onto another homeomorphic manifold in a ‘surrounding’? In other words embedding is homeomorphism except that the image is now put in a bigger environment?

Examples of topological Immersion which are not embedding are $mathbb{P}^2$ immersed in $mathbb{R}^3$ , on the part where the immersed $mathbb{P}^2$ intersects itself it is not 1-1 but locally 1-1.