Requirement "closed under finite intersection" in Van-Kampen-Theorem

Question

Given a topological space $X$, a point $x_0 \in X$ and an open cover $\mathcal{U}$ of $X$ of path-connected subsets containing $x_0$ which is closed under finite intersections, we have by Van-Kampen $$ \pi_1(X,x_0) = {\text{colim}}_{U \in \mathcal{U}} \bigl(\pi_1(U,x_0)\bigr). $$

Is there a counter example in which all requirements are satisfied except "closed under finite intersection"?

Answer 1

As a concrete and (to my mind at least) particularly bad example: Take $X = B(0,2) \subset \mathbb{R}^2$ the ball of radius 2 in the plane $U_1 = X - \{0\}$ and $U_2 = B(0,1)$. Then without accounting for the role of intersection, we would expect $\pi_1(B(0,2), (1/2,0)) \cong \mathbb{Z}$, while the full space is obviously contractible.

Answer 2

This would imply that every space covered by two open path connected sets $U_1,U_2$ containing $x_0$ would have a free product $\pi_1 U_1 * \pi_1 U_2$ as fundamental Group.

Answer 3

The 1984 paper available here gives the statement that if $\mathcal U =\{U_i,i \in I\} $ is a cover of $X$ by open sets and $A$ is a set which meets every path component of every 1-,2-,3-fold intersection of sets of $\mathcal U$ then the following diagram is a coequaliser of groupoids:

$$ \bigsqcup_{i,j}\pi_1(U_i\cap U_j,A) \rightrightarrows^a_b \bigsqcup_i \pi_1(U_i,A) \to ^c \pi_1(X,A) . $$ Here $c$ is induced by the inclusions $U_i \to X$ and $a,b$ are induced respectively by the inclusions $U_i \cap U_j \to U_i,U_i \cap U_j \to U_j$. The condition on $3$-fold intersections is necessary as shown by an example in the cited paper, and another example in Hatcher's book on Algebraic Topology for the case of groups, i.e. when $A$ is a singleton.

Note that a major point of the use of groupoids is to have a Seifert-van Kampen type theorem which yields the fundamental group of the circle, and of other unions of non pathconnected spaces. See this mathoverflow discussion.