Definition:Disconnected Set/Definition 3

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$ be a non-empty subset of $S$.

$H$ is a disconnected set of $T$ if and only if there exist non-empty subsets $U$ and $V$ of $H$ such that all of the following hold:

$H = U \cup V$

no limit point of $U$ is an element of $V$

no limit point of $V$ is an element of $U$.

Also see

• Equivalence of Definitions of Disconnected Set

• Results about disconnected sets can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): disconnected set
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): disconnected set