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I am just studying isometry in metric spaces.It is defined by a distance preserving bijection between the $2$ spaces.I think isometry means the $2$ spaces are essentially same as far as metric properties are concerned.But still,I am unaware of the power of the isometry.Can someone please help me to grasp the concept of isometry by providing a list of what thing can be done using isometry.I actually want to know how strong this isometry is.

Kishalay Sarkar
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  • This question might be useful for you. Isometrics are the isomorphisms in the category of metric spaces. It is worth to compare the situation with "isomorphisms of Groups" and "bijections of Sets." On the other hand, if you are looking for applications Functional analysis would be a rich source where you see many "isometric" constructions. – Bumblebee Feb 28 '20 at 02:36

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You pretty much have it: isometry preserves metric properties exactly---nothing more, nothing less. And so any generalization of metric properties (namely, topological properties) are also preserved.

EDIT: Maybe think of it this way. The term "isomorphism" means different things in different contexts. An isometry is just an isomophism w.r.t. metric properties. But due to the fact that metric spaces have topological properties, the term "isomorphism" is reserved for those, and a special term "isometry" is used for an isomorphism w.r.t. the specifically metric properties.

Ben W
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