A property of a topological space is called a homeomorphic property if it remains invariant under homeomorphisms
Is boundedness of a subspace of a metric space is a homeomorphic porperty?
For the first question, I can prove so far as to show that completeness is not a Topological property as if you consider (0,1), then one metric is the ordinary Euclidean metric on (0,1); a second metric is what you get when you pull the Euclidean metric on R back to (0,1) via a homeomorphism.
In one of those metrics, (0,1) is complete, but in the other it is not, though the two metrics generate the same topology. But how do I show go about showing that comepleteness remains invariant under homeomorphisms (or not)?
For the second question I am not sure where to begin as total boundedness is a not a topological property but I cannot say about boundedness in general. And it talks of metric space in the question, so this line of reasoning won't work at all I guess.