Fundamental Group is Independent of Base Point for Path-Connected Space

Theorem
Let $X$ be a path-connected space.

For $x \in X$ let $\pi_1(X,x)$ denote the fundamental group of closed paths with initial point $x$.

For $x, y \in X$, there is an isomorphism:
 * $\phi: \pi_1 \left({X, x}\right) \to \pi_1 \left({X, y}\right)$