Definition:Connected (Topology)/Topological Space

Definition

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

Definition 1

$T$ is connected if and only if it admits no separation.

Definition 2

$T$ is connected if and only if it has no two disjoint nonempty closed sets whose union is $S$.

Definition 3

$T$ is connected if and only if its only subsets whose boundary is empty are $S$ and $\O$.

Definition 4

$T$ is connected if and only if its only clopen sets are $S$ and $\O$.

Definition 5

$T$ is connected if and only if there are no two non-empty separated sets whose union is $S$.

Definition 6

$T$ is connected if and only if there exists no continuous surjection from $T$ onto a discrete two-point space.

Definition 7

$T$ is connected if and only if:

there do not exist disjoint, non-empty open sets $X$ and $Y$ of $T$ such that $X \cup Y = S$.

Also see

• Equivalence of Definitions of Connected Topological Space

• Results about connected spaces can be found here.