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
Since $T$ is connected, it admits no partition by Equivalence of Definitions of Topological Connectedness.

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 $t_2$.