Dense-in-itself Totally Explained

Everything about Dense-in-itself totally explained

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 
eq x

. On the other hand, this set of irrationals isn't 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.

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