Dense-in-itself

In <a href="/facts/General_topology/ezC7B0yP">general topology</a>, a subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> is said to be dense-in-itself or crowded
if 
 
 
 
 A
 
 
 {\displaystyle A}
 
 has no <a href="/facts/Isolated_point/d3EeWxuD">isolated point</a>.
Equivalently, 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is dense-in-itself if every point of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a <a href="/facts/Limit_point/95MykoGU">limit point</a> of 
 
 
 
 A
 
 
 {\displaystyle A}
 
.
Thus 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is dense-in-itself if and only if 
 
 
 
 A
 ⊆
 
 A
 ′
 
 
 
 {\displaystyle A\subseteq A'}
 
, where 
 
 
 
 
 A
 ′
 
 
 
 {\displaystyle A'}
 
 is the <a href="/facts/Derived_set_(mathematics)/SqeHgT2t">derived set</a> of 
 
 
 
 A
 
 
 {\displaystyle A}
 
.
A dense-in-itself <a href="/facts/Closed_set/upYnRTUj">closed set</a> is called a <a href="/facts/Perfect_set/wKUjfCkM">perfect set</a>. (In other words, a perfect set is a closed set without isolated point.)
The notion of <a href="/facts/Dense_set/nPT9Yxao">dense set</a> is distinct from dense-in-itself. This can sometimes be confusing, as "X is dense in X" (always true) is not the same as "X is dense-in-itself" (no isolated point).

Dense-in-itself open-in-new

Dense-in-itself