I want to compute the fundamental group of a space containing comb space along with boundary of $I \times I$ where I is $[0,1]$. Here is my attempt using transfinite induction and Van Kampen.
Let's call the given space $X$. Consider an open set $U_{1}$ say $(-1, 2/3) \times (-1, 2)$ and $U_{2}$ $(5/4, 2) \times (-1, 2)$. Then it's easily seen that hypothesis of Van Kampen Theorem is satisfied and $U_{1}$ $\cap$ $U_{2}$ is contractible while $U_{2}$ $\cap$ $X$ is homotopic equivalent to $S^{1}$ and $U_{1}$ $\cap$ $X$ is homotopic equivalent to one less vertical line in $X$. By induction hypothesis we assume that Fundamental group of $U_{1}$ $\cap$ $X$ is countable free product of $\mathbb{Z}$. Hence so is the Fundamental group of $X$.