Consider R with the standard metric. Let ECR be a subset which has no limit points. Show that E is at most countable.
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- If E is a subset of a metric space (X, d), show that E is nowhere-dense in X if and only if E c is dense in X.B. Given a metric space M with metric d, verify that any ε-ball is an open set.Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the Manhattan metric.Hint: As usual, there are two directions to prove in an ⇔.
- If S is a closed bounded subset of a metric space X, then S is compact.Let E be a subset of X, where X is a metric space. Show the limit points of E are the same as the limit points of the closure of E. That is, show E'=(E U E')'Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric.Hint: As usual, there are two directions to prove in an ⇔.
- Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric. Hint: As usual, there are two directions to prove in an ⇔.Let (X, d) be a metric space with X being a countable set. Show that X is not connected.Let (X, d) be a metric space with X being an infinite countable set. Show that X is not connected.