Final answer:
B(r, r), the set of bounded functions f: R -> R with bound r, is closed under the uniform topology because a uniform limit of functions within B(r, r) is also bounded by r. However, it is not closed under the topology of compact convergence, as a limit of functions that converges on compact sets may exceed the bound r outside of compact sets.
Step-by-step explanation:
The question is about showing that the set B(r, r) of bounded functions is closed in the uniform topology, but not in the topology of compact convergence. In the uniform topology, a sequence of functions {fn} in B(r, r) that converges uniformly to a function f implies that f is also a member of B(r, r), hence B(r, r) is closed in this topology.
However, B(r, r) is not closed in the topology of compact convergence because convergence on compact sets does not necessarily ensure convergence on the whole real line while maintaining boundedness. To illustrate this point, consider a sequence of functions that converge on all compact sets but the limit function exceeds the bound r outside a certain compact set, thus not belonging to B(r, r).