A subset is dense when every nonempty open set meets it, equivalently when its closure is the whole ambient space.
There are dense subsets of , where . For a fixed Cantor set of size , its complement is dense, and injects into the family of dense subsets. The reverse injection is the inclusion into . The Cantor-Schröder-Bernstein theorem gives equality.
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In mathematics, particularly in topology, a set \( A \) is referred to as a **dense set** in a space \( X \) if every point in \( X \) is either an element of \( A \) or is arbitrarily close to a point in \( A \). More formally, a subset \( A \) of a topological space \( X \) is dense in \( X \) if the closure of \( A \) is equal to \( X \).
