Hausdorff Metric: Definition 1 vs. Definition 2

[Mr Davis 97]

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.353.633&rep=rep1&type=pdf

The Hausdorff metric is defined on the space of nonempty closed bounded subsets of a metric space. The resulting metric space will be referred to as the "induced Hausdorff metric space", or else simply as the "induced Hausdorff space". Normally the term "Hausdorff space" refers to a space satisfying a certain topological separation axiom. But since this paper refers only to metric spaces,
which all satisfy the Hausdorff separation axiom, there will be no ambiguity.

<Moderator's note: Quotation added to make clear, what the author meant by "Hausdorff metric".>

The article above explores the notion of the Hausdorff metric. In the beginning of the article it describes how, if M is a metric space, the Hausdorff space induced by M is the set of all nonempty subsets of M that are closed and bounded together with the Hausdorff metric. Call this definition 1.

Here is my question: What if we instead claimed that the Hausdorff space induced M is the set of all nonempty compact subsets of M? Call this definition 2. Since the set of all nonempty compact subsets of M is a subset of the set of all nonempty subsets of M that are closed and bounded, do some of the results, which use definition 1, still follow if we use definition 2? For example, if I prove that definition 1 forms a metric space, is it immediately obvious that definition 2 does also, since it is a metric subspace of the former?

 

[martinbn]

Since the set of all nonempty compact subsets of M is a subset of the set of all nonempty subsets of M that are closed and bounded

M is a metric space. Why do you think that these two families of subsets are not the same?