Give an example of each of the following:
a) an infinite subset of R with no cluster point
b) a complete metric space that is bounded but not compact
c) a metric space none of whose closed balls is complete
a) We consider the integer set Z in R. Z has no cluster point.
Because for any integer n in Z, we consider its neighborhood (n - 0.5, n + 0.5). In this neighborhood, there does not exist another point in Z.
b) For the real set R, we consider the discrete metric d with d(x, y) = 1 if x is not equal to y and d(x, x) = 0.
Then R is ...
This solution explains how to solve the given problems involving Metric Spaces.