I wish to solve the following problem:
Let $D = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq 1\}$. Let $Y = \{(x, 0, 1/2) \in \mathbb{R}^3 \mid x \in \mathbb{R}\} \cup \{(0, 0, z) \in \mathbb{R}^3 \mid z \leq 1/2 \}$. Let $X = D \cap (\mathbb{R} \setminus Y)$, i.e., $X$ is a ball with a ``T'' removed. Calculate $\pi_1(X)$ and $H_*(X)$.
Intuitively, I can squish X around until it looks ''mostly'' (see below) like a solid double torus (a genus-2 surface unioned with its interior). I can compute $\pi_1$ and $H$ of this shape, but I have some questions about my method.
The original space $X$ is not closed. The ''T'' subtracted from the interior is an open set. So the original space does not include its boundary. Does this affect $\pi_1$ or $H_*$ in any way? Why?
Squishing stuff around in my head seems like a dangerous way to do math. Is there a more rigorous way to approach this problem? I can find decompositions that will let me use Meyer-Vietoris, but I can't find a good decomposition for the Van Kampen theorem.