# Thread: Nowhere dense in a metric space

1. ## Nowhere dense in a metric space

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 $\bar{A}$ 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.

2. 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 $\bar{A}$ 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.
NOTE: You may want to wait for a more senior knowledgable member to either validate or correct my suggestion.

This is almost by defintion. Supose the opposite that $A$ is dense in $X$ and that the hypothesis holds...use this to show that not every point of $X$ is a limit point of $A$

3. Originally Posted by aliceinwonderland
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.
Is it still true if the word "closure" is taken out such 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 which is disjoint from A"
If it is wrong, any counterexample?