Connected Space is Connected Between Two Points

Theorem

Let $T$ be a topological space which is connected.

Then $T$ is connected between two points.

Proof

By definition of connected space, $T$ admits no separation.

Therefore, vacuously, every partition has one open containing $t_1, t_2 \in T$, for all $t_1, t_2 \in T$.

That is, for all $t_1, t_2 \in T$, $T$ is connected between $t_1$ and $t_2$.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness