A connected, but not path-connected, space whose fundamental group depends on the basepoint

Question

It is a well known result that the fundamental group of a path-connected space is independent (up to isomorphism) of the choice of the basepoint. Can someone provide an explicit example of a connected, but not path-connected, space for which the fundamental group does indeed depend on the basepoint?

Answer 1

Let $X = \{(\sin(t), \cos(t), \arctan(t)) \in \mathbb{R}^3 \mid t \in \mathbb{R}\}$, then let $Y = S^1 \times \{-\pi/4\}$ and $Z = (S^1 \times \{\pi/4\})\vee W$ for any path-connected space $W$ with non-trivial fundamental group.

Now take the space $X \cup Y \cup Z$ which is connected ($X$ gets arbitrarily close to both $Y$ and $Z$ and so cannot be separated from either by open neighbourhoods), but has three distinct path-components, $X$, $Y$ and $Z$, with $\pi_1(X) = 1$, $\pi_1(Y) = \mathbb{Z}$ and $\pi_1(Z) =\mathbb{Z} \ast \pi_1(W)$ which are each non-isomorphic.