This is a different approach to what was suggested by Joel Pereira in my question Fundamental group of theta-space and the doubly punctured theta space in Munkres Topology Example 70.1
Munkres Topology Example 70.1
Let X be theta-space, U = $X \setminus \{a\}$ and V = $X \setminus \{b\}$. Let $U \cap V = X \setminus \{a,b\}$ be doubly punctured theta-space where $a,b$ are interior points of $A$ and $B$.
The doubly punctured space $\mathbb R^2 \setminus \{p,q\}$ has deformation retracts Figure 8 and $X$. I think the Figure 8 for $p=(-1,0)$ and $q=(1,0)$ is or can be 2 circles centered at $p$ and $q$ that intersect only at the origin $(0,0)$. If Figure 8 has the same homotopy type as $\{(0,0)\}$ and $X$ has the same homotopy type as $X \setminus \{a,b\}$, then $X \setminus \{a,b\}$ has the same homotopy type as $\{(0,0)\}$.
Is this argument (not necessarily its assumptions!) right?
Does Figure 8 have the same homotopy type as $\{(0,0)\}$?
Does $X$ have the same homotopy type as $X \setminus \{a,b\}$?
If the answers are all yes, I can work out the details on my own, and you can just say so in an answer without explanation and then I'll accept your answer.
If the answer to 1 is no, then why?
If the answer to 1 is yes but the answer to 2 or 3 is no, then again you can just say so in an answer without explanation and then I'll accept your answer.