This question is related to my previous question: Question about a statement: why taking linear span?
There the answer was satisfactory but I am wondering now about some examples of tangent sets that are not linear or not closed or both. I will recall here the definition of tangent set. Before we need:
Definition A map $\psi: \mathcal{P} \rightarrow \mathbb{B}$ is differentiable at $P$ relative to a given tangent set $\dot{\mathcal{P}}_P$ if there exists a continuous linear map $\dot{\psi}_P: L_2(P) \rightarrow \mathbb{B}$ such that for every $g \in \dot{\mathcal{P}}_P$ and a submodel $t \rightarrow P_t$ with score function $g$, $$ \frac{\psi\left(P_t\right)-\psi(P)}{t} \rightarrow \dot{\psi}_P g $$
In words we say that a differentiable path is a parametric submodel $\left\{P_t: 0 \leq t<\varepsilon\right\}$ that is differentiable in quadratic mean at $t=0$ with score function $g$. Letting $t \rightarrow P_t$ range over a collection of submodels, we obtain a collection of score functions, which we call a tangent set of the model $\mathcal{P}$ at $P$, and denote by $\dot{\mathcal{P}}_P$. Moreover every score function satisfies $P g=0$ and $P g^2<\infty$.
It is really hard for me to think about some examples since I am a beginner in this field.