Template:Compact topological surfaces - English Wikipedia @ Freddythechick

Compact topological surfaces and their immersions in 3D

Without boundary

Orientable	Sphere (genus 0) Torus (genus 1) Number 8 (genus 2) Pretzel (genus 3) ...
Non-orientable	Real projective plane genus 1; Boy's surface Roman surface Klein bottle (genus 2) Dyck's surface (genus 3) ...

With boundary

Related
notions

Properties	Connectedness Compactness Triangulatedness or smoothness Orientability
Characteristics	Number of boundary components Genus Euler characteristic
Operations	Connected sum Making a hole Gluing a handle Gluing a cross-cap Immersion