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≠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.
Title | dense in-itself |
---|---|
Canonical name | DenseInitself |
Date of creation | 2013-03-22 14:38:29 |
Last modified on | 2013-03-22 14:38:29 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 4 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 54A99 |
Related topic | ScatteredSpace |