The empty set is nowhere dense. In a discrete space, the empty set is the only such subset. In a T1 space, any singleton set that is not an isolated point is nowhere dense. The boundary of every open set and of every closed set is nowhere dense.

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## Which of the following sets are nowhere dense?

The empty set is nowhere dense. In a discrete space, the empty set is the only such subset. In a T1 space, any singleton set that is not an isolated point is nowhere dense. The boundary of every open set and of every closed set is nowhere dense.

**Are the Irrationals nowhere dense?**

No they are not: Wikipedia and Wolfram MathWorld indicate that a “nowhere dense set” is one whose closure has empty interior.

**Is the complement of a dense set nowhere dense?**

Otherwise put, A is nowhere dense iff it is contained in a closed set with empty interior. Passing to complements, we can say equivalently that A is nowhere dense iff its complement contains a dense open set (why?). Proposition 7.1 Let X be a metric space. Then: (a) Any subset of a nowhere dense set is nowhere dense.

### Is nowhere dense set closed?

A nowhere dense set is not necessarily closed, but the closure of a nowhere dense set is still nowhere dense, and is of course closed.

**Is the empty set nowhere dense?**

(∅−)∘=∅ and so by definition ∅ is nowhere dense in T.

**Is Q nowhere dense in R?**

For example, Z is nowhere dense in 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)⊂ˉZ=Z. An example of a set which is not dense, but which fails to be nowhere dense would be {x∈Q|0

#### Is rationals dense set?

The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself.

**Is the Cantor set closed?**

Cantor set is a special subset of the closed interval [0,1] invented by a German mathematician Georg Cantor in 1883. We have already dicussed the construction of this ‘ternary’ set in the class but let me quickly recall it.

**Is RA dense set?**

Thus, R α R_{\alpha} Rα is dense in [ 0 , 1 ] [0,1] [0,1].

## Is N dense in R?

Because (0,1) is an open set, it intersects any dense subset of R. This implies that N is not dense in R, as it does not intersect (0,1).

**What does it mean if a set is nowhere dense?**

For example, Z is nowhere dense in 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. An example of a set which is not dense, but which fails to be nowhere dense would be { x ∈ Q | 0 < x < 1 }.

**What is an example of a set that is not dense?**

For example, Z is nowhere dense in 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. An example of a set which is not dense, but which fails to be nowhere dense would be { x ∈ Q | 0 < x < 1 }. Its closure is [ 0, 1], which contains the open interval ( 0, 1).

### What is an example of nowhere dense in R?

For example, Z is nowhere dense in 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) ⊂ ˉZ = Z. An example of a set which is not dense, but which fails to be nowhere dense would be {x ∈ Q | 0 < x < 1}. Its closure is [0, 1], which contains the open interval (0, 1).

**How do you know if a set is dense?**

A set might not be dense, but it might be the case that it is dense locally according to a suitable part of the whole space; so one can consider the following definition: A subset Y ⊆ X is called to be somewhere dense if there exists a non-empty open set U ⊆ X such that we have Y ∩ U ¯ = U ¯.