Show that

A subsetAof 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 fromA.

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Some definitions

1. A subsetAof a metric space X is nowhere dense in X if has empty interior.

2. LetAbe a subset of a metric space X. A point x inAis an interior point ofAprovided that there is an open set O which contains x and is contained inA.