Topological Space with Generic Point is Path-Connected

Theorem
Let $X$ be a topological space.

Let $X$ have a generic point $g \in X$.

Then $X$ is path-connected.

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

Let $x\in X$.

Define a path $\gamma : [0 .. 1] \to X$ by:
 * $\gamma(t) = \begin{cases}

x & : t \leq \frac12 \\ g & : t > \frac 12 \end{cases}$

We show that $\gamma$ is indeed continuous.

Let $U \subseteq X$ be open and nonempty.

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

If $ x \in U$, then its preimage $\gamma^{-1}(U) = [0 .. 1]$ is open.

If $x \notin U$, then its preimage $\gamma^{-1}(U) = (\tfrac12 .. 1]$ is also open.

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