An isomorphism is a bijective mapping between two spaces, which preserves the structure of and allowed operations in the space. Under different scenarios, the said structure can be different, for example:

  • Metric space: the metric, i.e. distance → isometry
  • Group: multiplication → bijective group homomorphism
  • Vector space: addition of vectors and product of a scalar value and a vector
  • Topological space: openness of the sets → homeomorphism
  • In differential geometry
    • When differentiability is appended to the forward mapping → differentiable homeomorphism
    • When differentiability is appended to both forward and inverse mapping → diffeomorphism

With this concept, two spaces can be identified in the sense that operations on corresponding points in these two spaces lead to same effects, while we do not care the exact or numerical representation of a point in these spaces. Of course, there is no need to do this: once the features of one space have been studied and clarified, the other space is automatically known to us.