Question

Difference between homeomorphism and isomorphism

Answer 1

Homomorphism - an algebraical term for a function preserving some algebraic operations. For a group homomorphism ϕϕ we have ϕ(ab)=ϕ(a)ϕ(b)ϕ(ab)=ϕ(a)ϕ(b) and ϕ(1)=1ϕ(1)=1, for a ring homomorphism we have additionally ϕ(a+b)=ϕ(a)+ϕ(b)ϕ(a+b)=ϕ(a)+ϕ(b) and for a vector-space homomorphism also ϕ(r⋅a)=r⋅ϕ(a)ϕ(r⋅a)=r⋅ϕ(a), where rr is a scalar and aa is a vector.

Isomorphism (in a narrow/algebraic sense) - a homomorphism which is 1-1 and onto. In other words: a homomorphism which has an inverse.