Question

A set A, a subset of the metric space (X,d), is said to be nowhere dense if A0=∅. It can be shown (see Exercise 1) that this is equivalent to requiring that CAˉ be dense in X or that CAˉ=X. REMARK I. If d is the trivial metric, the only nowhere-dense set is the n

Answer 1

A set A, a subset of a metric space (X,d), is said to be nowhere dense if its closure has an empty interior or if its complement is dense in X.

Step-by-step explanation:

In the context of metric spaces, a set A that is a subset of the metric space (X,d) is said to be nowhere dense if the closure of the set A, denoted by A-bar, has an empty interior (A0=∅). This means that there are no open sets in X contained completely within A-bar.

Alternatively, a set A is nowhere dense if its complement, denoted by CA, is dense in X or if CA=X. In other words, there are points in X arbitrarily close to every point in the complement of A.

For example, in the real numbers with the standard metric, the set of rational numbers is nowhere dense since its closure is the entire real line, while the set of irrationals has an empty interior and is also nowhere dense.