Topological Space with Generic Point is Path-Connected

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $T$ have a generic point $g \in S$.

Then $T$ is path-connected.

Proof

By Path-Connectedness is Equivalence Relation, it suffices to prove that every point is path-connected with $g$.

Let $x \in S$.

Define a path $\gamma: \closedint 0 1 \to S$ by:

$\map \gamma t = \begin{cases}

x & : t \le \dfrac 1 2 \\ g & : t > \dfrac 1 2 \end{cases}$

We show that $\gamma$ is indeed continuous.

Let $U \subseteq S$ be open and non-empty.

Because $g$ is a generic point, $g \in U$.

If $x \in U$, then its preimage $\gamma^{-1} \sqbrk U = \closedint 0 1$ is open.

If $x \notin U$, then its preimage $\gamma^{-1} \sqbrk U = \hointl {\dfrac 1 2} 1$ is also open.

Thus $x$ and $g$ are path-connected.

$\blacksquare$