Answer to Question #217024 in Differential Geometry | Topology for Prathibha Rose

Question

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

Answer 1

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