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学得有点杂，这些概念其实都有 cover，这里总结一下：

f:X→Y X & Y remark explanation
bijection sets    
isomorphism algebra structures structure-preserving E.g. if X is a field, then Y must be a field
homeomorphism topological spaces topological structure-preserving Y must preserve all topological properties that X has
isometry metric spaces distance-perserving ∀a,b∈X:dY(f(a),f(b))=dX(a,b)

网友 ziggurism 的总结:

  • If there is a one-to-one and onto map between two sets, it is invertible, and we say it is a bijection. For the purposes of set theory, if there is a bijection between two sets, they may be treated as the same set.
  • If there is a one-to-one and onto algebra preserving map of groups, rings, fields, etc, whose inverse also preserves the algebraic structure, we say it is an isomorphism. For the purposes of group theory (ring theory, etc), if there is a isomorphism between two groups, they may be treated as the same group.
  • If there is a one-to-one and onto mapping of topological spaces which preserves the topological data, and whose inverse also preserves the topological data, we say it is a homeomorphism. For the purposes of topology, if there is a homeomorphism between two spaces, they may be treated as the same space.
  • If there is a one-to-one and onto mapping of metric spaces which preserves the distance relation, and whose inverse also preserves the distance relation, we say it is an isometry. For the purposes of metric topology, if there is an isometry between two spaces, they may be treated as the same space.

网友 rogerl 的观点:

If you are talking just about sets, with no structure, the two concepts (bijection vs isomorphism) are identical.

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