## You are here

Homedense in-itself

## Primary tabs

# dense in-itself

A subset $A$ of a topological space is said to be *dense-in-itself* if $A$ contains no isolated points.

Note that if the subset $A$ is also a closed set, then $A$ will be a perfect set. Conversely, every perfect set is dense-in-itself.

A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers. This set is dense-in-itself because every neighborhood of an irrational number $x$ contains at least one other irrational number $y\neq x$. On the other hand, this set of irrationals is not closed because every rational number lies in its closure.

For similar reasons, the set of rational numbers is also dense-in-itself but not closed.

Related:

ScatteredSpace

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

54A99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff