Nit (taking "homomorphism" to mean the more general notion of a morphism where you just preserve some property): an isomorphism is an invertible homomorphism wh

Nit (taking "homomorphism" to mean the more general notion of a morphism where you just preserve some property): an isomorphism is an invertible homomorphism where the inverse is also a homomorphism. For some structures (e.g. vector spaces+linear maps, and I believe all other algebraic structures), the inverse of a homomorphism is always also a homomorphism. For others (e.g. along the lines of your connectivity example, topological spaces+continuous maps), the inverse does not necessarily preserve the structure you are interested in.