# Dense-In-Itself

Tech

## What is Dense-in-itself?

In mathematics, a subset of a topological space is said to be dense-in-itself if contains no isolated points. Every dense-in-itself closed set is perfect. 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 (considered as a subset of the real numbers). This set is dense-in-itself because every neighborhood of an irrational number contains at least one other irrational number . 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 (also considered as a subset of the real numbers) is also dense-in-itself but not closed. The above examples, the irrationals and the rationals, are also dense sets in their topological space, namely . As an example that is dense-in-itself but not dense in its topological space, consider . This set is not dense in but is dense-in-itself.

topology

### Translations

Zbiór wszędzie gęsty (pl)مجموعة مكثفة في حد ذاتها (ar)자기 조밀 공간 (ko)

## Tech Info

Source: [object Object]
— Date merged: 11/6/2021, 1:32:44 PM
— Date scraped: 5/20/2021, 5:52:23 PM