# 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.

## Also see

- Results about
**connected spaces**can be found here.