Question

These problems concern the discrete metric. You can assume that the underlying space is R. (a)...

These problems concern the discrete metric. You can assume that the underlying space is R.

(a) What does a convergent sequence look like in the discrete metric?

(b) Show that the discrete metric yields a counterexample to the claim that every bounded sequence has a convergent subsequence.

(c) What does an open set look like in the discrete metric? A closed set?

(d) What does a (sequentially or topologically) compact set look like in the discrete metric?

(e) Show that the discrete metric yields a counterexample to the claim that every closed and bounded set is compact.

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