Let $X\subset\mathbb{P}^2$ be a path connected topological space. Suppose we have path connected closed sets $A,A_1,A_2,\cdots,A_m\subset X$ such that $X=A\cup_{i=1}^m A_i$ where
a) each $A\cap A_j$ is non-empty path connected and a neighbourhood deformation retract in each of $A_i$ and $A$,
b) $A_i\cap A_j=\emptyset$ for $i\neq j$,
c) $A_i\cap A_j\cap A_k=\emptyset$ for $i\neq j\neq k$.
Is there a version of Van-Kampen theorem that applies to this. The problem is that there is no base point common to all the sets.
Alternately, I could take $B=\cup_{i}^m A_i$ and $X=A\cup B$ where $A\cap B$ is a union of finitely many path connected spaces (Infact I know that $A\cap B$ is a union of finitely many circles). In this case can I say something about $\pi_1(X)$ in terms of $\pi_1(A)$ and $\pi_1(B)$?
I would be grateful for any reference.
This is another question which shows the advantage of moving from groups to groupoids, with the notion of the fundamental groupoid $\pi_1(X,S)$ on a set $S$ of base points, chosen according to the geometry. In the case presented, choose $S$ to consist of one point $s_i$ in each $A \cap A_i$. For more see the books Topology and Groupoids or Categories and Groupoids. The second book has more on the use of the algebra of groupoids. The first editions of these books were published in $1968$ and $1971$ respectively. Essentially, you are computing $\pi_1(X,S)$ for $X$ given as a succession of adjunction space $A \cup_{f_i} A_i $ where $f_i: A \cap A_i \to A$.
Thus the "natural" answer is in terms of groupoids. Compare this mathoverflow discussion.
Of course group theory has been a major part of mathematics and its applications for more than two centuries, but the extension of the realm of discourse to groupoids is useful for giving algebraic models in topology and has many applications, including "higher dimensional groupoids", which get more complicated with increasing dimension.