Definition:Connected (Topology)/Topological Space

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Definition

Let $T = \left({S, \tau}\right)$ 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 $\varnothing$.


Definition 4

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


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.


Also see

  • Results about connected spaces can be found here.