Suppose I have a topological space $\Delta$, a point $v \in \Delta$, and $\pi_1(\Delta, v) = G$ for some group $G$. Does there exist a "minimal subset" $S$ of $\Delta$ containing $v$ such that $\pi_1(S, v) = \pi_1(\Delta, v)$?
Here was my idea:
We introduce the partial order $\leq$ on the set:
\begin{align} \mathcal{S} := \{ S \ | \ v \in S \subseteq \Delta \ \text{and} \ \pi_1(S,v) = \pi_1(\Delta,v) \} \end{align}
by $A \leq B \iff A \subseteq B$. We define the dual order $\leq^*$ on $\mathcal{S}$ by $A \leq^* B \iff B \leq A$. I want to apply Zorn's lemma to $\mathcal{S}$ by stating that every chain has a maximal element with respect to $\leq^*$, but I'm having trouble finding what this element should be. Obviously if $\Delta$ has trivial fundamental group, this maximal element would universally just be the singleton $\{ v \}$, but what about more complicated spaces? Is this even possible? If not, are there any additional restrictions we can place on $\Delta$ that lets us find such a subset?