In theta space, is the lower arc and line segment a deformation retract of punctured theta?

Question

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 $B$.

Is $B \cup C$ a deformation retract of $U$?

Answer 1

I think it's a retract by either $r(z)=z1_{B \cup C} + Re(z)1_{C \cup [A \setminus \{a\}]}$ or $r(z)=z1_{B \cup C} + \overline{z}1_{C \cup [A \setminus \{a\}]}$ either of which is continuous by the pasting lemma because $B \cup C$ and $C \cup [A \setminus \{a\}]$ are closed in $U$ because

$$B \cup C = U \cap \{Im(z) \le 0 \}$$

$$C \cup [A \setminus \{a\}] = U \cap \{Im(z) \ge 0 \}$$

I think it's a deformation retract with the straight line homotopy $H(z,t)=(1-t)z+tr(z)$ because:

1. $H(z,0)=z \ \forall z \in B \cup C$
2. $H(z,1)=r(z) \ \forall z \in B \cup C$
3. $H(d,t)=(1-t)(d)+tr(d)=d \ \forall \ d \in B \cup C, t \in I$