for further examples, if you take a surface of positive genus and "stretch the neck" of some embedded circle like before, you can send part of it off to infinity, and in that way decrease genus. eg genus 2 surface, stretche the connected sum circle into a long cylinder, leaving one of the tori at the origin and letting the other go to infinity. Then you limit to a properly embedded punctured torus. so that's how one can go between things of positive genus without just going to the empty set