Question #217024

Give an example of a nowhere dense set in a metric space .substantiate your claim


1
Expert's answer
2021-07-19T08:10:27-0400

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, Z\mathbb{Z} is nowhere dense in R\mathbb{R} because it is its own closure, and it does not contain any open intervals (i.e. there is no (a, b) s.t. (a,b)Z=Z(a, b) \subset \overline{\mathbb{Z}}=\mathbb{Z} . An example of a set which is not dense, but which fails to be nowhere dense would be {xQ0<x<1}\{x \in \mathbb{Q} \mid 0<x<1\} . 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 (0,1)R(0,1) \subset \mathbb{R} .


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