I'm exploring possible examples for metric spaces, examples of interest: for example
- Fix set, vary distance function. Possible changes were studied. Real with usual and discrete metric is an example.
- Fix a metric, vary sets. For example, Real and a subset with same metric.
Second case led me to two different questions, which is my doubt here:
- Can we have a case where A and B are metric spaces with same metric, but not subset of each other (like real line and a subset with same distance function.)
- Can we have a case where a distance function is a metric on a set A, but not in a set B?
May be the arguments are trivial, but I can't make any good progress on this matter.
It is a bit unclear what exactly you mean by "same metric" and "same distance function" when the domains of those functions are not the same, so the functions themselves cannot be the same. I'll assume you mean "given by the same formula".
For $(2)$, consider the power set of a set X, with $d : \mathcal{P}(X) \times \mathcal{P}(X) \to \mathbb{R}$ given as $d(A, B) = |(A \cup B)\setminus (A \cap B)|, \forall A, B \in \mathcal{P}(X)$ (the cardinality of the symmetric difference of $A$ and $B$). You can check that $d$ is a metric on $\mathcal{P}(X)$ if $X$ is finite, and otherwise it is not as $d$ can be infinite. Notice that this also provides an example for $(1)$: take the power sets of two disjoint sets with this metric $d$.
A similar example for (2): define a distance between two real valued functions $f, g : D \to \mathbb{R}$ as $d(f, g) = \sup\limits_{x\in D} |f(x) - g(x)|$. Now $d$ is a metric on $C[a, b]$, but not on $C(\mathbb{R})$, for the same reason as above.