Typical equivalence relations between mathematical spaces

• Bijection: this is the equivalence in the sense of one-to-one and surjective mapping or correspondence between two sets.
• Isomorphism: this is the equivalence in the sense of algebraic structure of groups.
• Homeomorphism: this is the equivalence in the sense of topology and requires the forward map \(f\) and backward map \(f^{-1}\) to be continuous. It includes bijection.
• Isometry: this is the equivalence in the sense of metric.
• Differentiable homeomorphism: it is a homeomorphism with the forward map \(f\) being differentiable.
• Diffeomorphism: it is a homeomorphism with both the forward map \(f\) and backward map \(f^{-1}\) being differentiable.