Give an example of a connected space X with two points $x_0$ and $x_1$ such that $\pi_1(X,x_0)$ is not isomorphic to $\pi_1(X,x_1)$

Question

In preparation for an upcoming test I have been looking through old tests and found this question:

Give an example of a connected space X with two points $x_0$ and $x_1$ such that $\pi_1(X,x_0)$ is not isomorphic to $\pi_1(X,x_1)$?

I can't seem to find an example, is that because there isn't one?

Answer 1

Let $S$ denote the topologist sine curve, which is a union of a sine graph and an interval. Adjoin to this interval a circle. The points $x$ on the interval and on the circle have $\pi(X,x)=\mathbb Z$, while those points $x'$ on the sine graph have $\pi(X,x') = 0$.

Answer 2

As a variation on the topologist's sine curve example, I will construct a "topologist's helix":

$$X \subseteq \mathbb{R}^3, X := \left\{ \left(\cos t, \sin t, \frac{1}{t} \right) \mid t > 0 \right\} \cup \{ (\cos t, \sin t, 0) \mid t \in [0, 2 \pi) \}.$$

The picture is of a helix winding around the cylinder $x^2 + y^2 = 1$, with the $z$ component approaching 0 more and more closely with each winding, unioned with the unit circle in the $xy$-plane.

The two parts I described above are the path-connected components of $X$. Now, the first component is contractible so $\pi_1(X, x) = 0$ for $x$ in this component; whereas the second component is clearly homeomorphic to $S^1$ so $\pi_1(X, x) = \mathbb{Z}$ for $x$ in this component.