A simple example of a set that is dense-in-itself but not closed (and hence not a perfect set) is the set of irrational numbers (considered as a subset of the real numbers). This set is dense-in-itself because every neighborhood of an irrational number x {\displaystyle x} contains at least one other irrational number y ≠ x {\displaystyle y\neq x} . On the other hand, the set of irrationals is not closed because every rational number lies in its closure. Similarly, the set of rational numbers is also dense-in-itself but not closed in the space of real numbers.
The above examples, the irrationals and the rationals, are also dense sets in their topological space, namely R {\displaystyle \mathbb {R} } . As an example that is dense-in-itself but not dense in its topological space, consider Q ∩ [ 0 , 1 ] {\displaystyle \mathbb {Q} \cap [0,1]} . This set is not dense in R {\displaystyle \mathbb {R} } but is dense-in-itself.
A singleton subset of a space X {\displaystyle X} can never be dense-in-itself, because its unique point is isolated in it.
The dense-in-itself subsets of any space are closed under unions.5 In a dense-in-itself space, they include all open sets.6 In a dense-in-itself T1 space they include all dense sets.7 However, spaces that are not T1 may have dense subsets that are not dense-in-itself: for example in the dense-in-itself space X = { a , b } {\displaystyle X=\{a,b\}} with the indiscrete topology, the set A = { a } {\displaystyle A=\{a\}} is dense, but is not dense-in-itself.
The closure of any dense-in-itself set is a perfect set.8
In general, the intersection of two dense-in-itself sets is not dense-in-itself. But the intersection of a dense-in-itself set and an open set is dense-in-itself.
This article incorporates material from Dense in-itself on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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