Definition:Connected (Topology)/Set/Definition 4

Definition

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

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

$H$ is a connected set of $T$ if and only if:

there do not exist disjoint, non-empty subsets $X$ and $Y$ of $H$ such that $X \cup Y = H$ such that:

no limit point of $X$ is an element of $Y$

no limit point of $Y$ is an element of $X$.

Also see

• Equivalence of Definitions of Connected Set

• Results about connected sets can be found here.

Sources

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