Path-Connected Space is Connected/Proof 1

Proof
Suppose $X \mid Y$ is a separation of $T$.

As $X$ and $Y$ are non-empty, we can find $x \in X$ and $y \in Y$.

As $T$ is path-connected, there exists a path $f : \closedint 0 1 \to T$ with initial point $x$ and final point $y$.

Subset of Real Numbers is Interval iff Connected shows that $\closedint 0 1$ is connected.

Continuous Image of Connected Space is Connected shows that $\Img f$ is connected.

Connected Subspace Lie in One Component of Separation shows that either $\Img f \cap X = \O$, or $\Img f \cap Y = \O$.

As we have $x, y \in \Img f$, this is a contradiction.

It follows that there can be no separation of $T$, so $T$ is connected.