Definition:Connected (Topology)/Topological Space
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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:
Also see
- Results about connected spaces can be found here.