Terminology: "Metric Properties ?" "Topological Properties" is a generally used term to refer to properties of a topological space which are preserved by homeomorphism. Can one use the term "metric properties" to refer to the non-topological properties of boundedness, Cauchy convergence, etc which are preserved by isometry ?
 A: I don't believe there is a universally agreed upon name for what you're describing.  It's worth noting that there are two other common types of homeomorphism lying between isometries and homeomorphisms.  We have the following hierarchy.


*

*Homeomorphisms - bijective maps continuous in both directions

*Uniform homeomorphisms - bijective maps uniformly continuous in both directions

*Bi-Lipschitz homeomorphisms - bijective maps Lipschitz continuous in both directions

*Isometries - bijective maps that preserve distance


Each of these preserve different things.  Homeomorphisms preserve topological properties, and  uniform homeomorphisms preserve uniform properties such as completeness, sequences being Cauchy, and total boundedness.  Isometries clearly preserve everything, so that leaves only bi-Lipschitz homeomorphisms.  We could call properties preserved by these "metric properties" or "bi-Lipschitz" properties.  The main property preserved here and not by uniform homeomorphisms that comes to mind is boundedness.  
My impression is that most people would understand what you meant if you said "metric properties" or "non-topological metric properties."
