Equivalence of Definitions of Connected Topological Space

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

$(1) \iff (2)$: No Separation iff No Union of Closed Sets
=== $(2) \implies (3)$: No Union of Closed Sets implies No Subsets with Empty Boundary ===

$(4) \implies (5)$: No Clopen Sets implies No Union of Separated Sets
=== $(5) \implies (6)$: No Union of Separated Sets implies No Surjection to Discrete Two-Point Space ===

=== $(6) \implies (1)$: No Surjection to Discrete Two-Point Space implies No Separation ===

Also see

 * Condition on Connectedness by Clopen Sets for a separate proof that $(1)$ and $(4)$ are equivalent.