Statistical Manifold with Non-trivial Topology

Let $(\Omega, E)$ be a measure space. An $n$-dimensional statistical model is then a tuple $(\Theta, \mathcal{M}, \Phi)$ where $\Theta \subseteq \mathbb{R}^n$ open, $\mathcal{M} = \{p_\theta := p(\cdot | \theta), \theta \in \Theta\}$ is the set of parameterized probability distributions on $\Omega$ and the mapping $\Phi: \Theta \to \mathcal{M}$ via $\theta \mapsto p(\cdot | \theta)$ is injective.$^1$

In this context, one refers to $\mathcal{M}$ as a statistical manifold, as the mapping $\Phi$ serves as a coordinate system for $\mathcal{M}$.

Question: Can anyone give me an example of a statistical manifold with non-trivial topology? If this is not possible, can you give an explanation for why this is.

Many thanks!

$^1$ This description is taken from here.