dense in-itself

A subset A of a topological spaceMathworldPlanetmath is said to be dense-in-itself if A contains no isolated pointsMathworldPlanetmath.

Note that if the subset A is also a closed setPlanetmathPlanetmath, then A will be a perfect setMathworldPlanetmath. 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 neighborhoodMathworldPlanetmathPlanetmath of an irrational number x contains at least one other irrational number yx. On the other hand, this set of irrationals is not closed because every rational number lies in its closureMathworldPlanetmathPlanetmath.

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