Give an example of a nowhere dense set in a metric space .substantiate your claim
Solution:
Let X be a metric space. A subset A ⊆ X is called nowhere dense in X if the interior of the closure of A is empty, i.e. (A)◦ = ∅. Otherwise put, A is nowhere dense iff it is contained in a closed set with empty interior.
For example, is nowhere dense in because it is its own closure, and it does not contain any open intervals (i.e. there is no (a, b) s.t. . An example of a set which is not dense, but which fails to be nowhere dense would be . Its closure is [0,1], which contains the open interval (0,1). Using the alternate definition, you can note that the set is dense in .
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