Exercises relate to the following definition: Let (X, d) be a metric space. A subset D of X is said to be dense in X if cl D=X.
Let D be a subset of a metric space (X, d).
(a) Prove that D is dense in X iff every nonempty open subset of X has a nonempty intersection with D.
(b) Prove that D is dense in X iff for every x ∈X and every ε > 0 there exists a point z in D such that d(x, z) < ε.
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