Topology

  • space, connectivity, continuity, near/far(order)
  • can be specified with the addition of operators: topology> metric > norm > inner product

Algebra 

  • finitary manipulation, equalities, no order

homomorphism preserves topology, isomorphism preserves algebra. (iaht)

isomorphism = homomorphism + bijective

homomorphism = algebra structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces while

homeomorphism = topologic continuous bijection from one topological space to another, with continuous inverse.

on topological space, isomorphism = homeomorphism

compact Hausdorff topological spaces: inverses are always continuous(~algebra inverses of homomorphisms = homomorphisms) and therefore checking the continuity of f suffices to show the isomorphism