Originally Posted by

**aliceinwonderland** Show that

A subset *A *of a metric space X is nowhere dense in X if and only if each non-empty open set in X contains an open ball whose closure is disjoint from *A*.

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Some definitions

1. A subset *A* of a metric space X is nowhere dense in X if $\displaystyle \bar{A}$ has empty interior.

2. Let *A *be a subset of a metric space X. A point x in *A *is an interior point of *A *provided that there is an open set O which contains x and is contained in *A.*