Finite wedges of quotient circles have van Kampen covers at the wedge point

The tagged quotient for $W_r\vee Q$ differs from that for $W_{r+1}$ only by grouping the old tagged summands; the quotient maps in both directions preserve every tag and are continuous by [F2], so they are inverse homeomorphisms. Under this identification, define $A_r$ to contain all of the old wedge $W_r$ and the arc $O$ in the new circle. Define $B_r$ to contain the whole new circle and the arc $O$ in every old circle. Their inverse images under the wedge quotient are open, by [F1] and the disjoint-union topology, and each inverse image is saturated because every listed arc contains its basepoint. Hence $A_r$ and $B_r$ are open; they cover $W_{r+1}$, and their intersection consists exactly of one copy of $O$ in every circle, with all their basepoints identified.

Use the coordinate $t\in (-1\mathrm{/}4,1\mathrm{/}4)$ supplied by [F1] and the contraction $t\mapsto (1-s)t$. On $A_r$, contract only the new-circle arc and fix $W_r$; on $B_r$, contract every old-circle arc and fix the new circle. The formulas agree at the tagged basepoints and are jointly continuous away from the wedge point. At the wedge point, a target neighbourhood contains an arc $\mathrm{\mid }t\mathrm{\mid }<{\epsilon }_j$ in every incident branch; there are finitely many branches, so their minimum is positive, and the contraction never increases $\mathrm{\mid }t\mathrm{\mid }$. Together with the fixed trace on the retract, this gives a product neighbourhood mapped into the target neighbourhood, proving joint continuity there. Thus the formulas are deformation retractions as in [F3], and [F5] makes the two retract inclusions induce fundamental-group isomorphisms.

Apply the same contraction simultaneously on every arc of $A_r\cap B_r$. Joint continuity away from $w$ is coordinatewise, and at $w$ the same finite-minimum neighbourhood argument from step 2.1 applies to all incident arcs. This gives a deformation retraction of the overlap onto $w$. The construction also covers $r=0$, when $W_0$ is the point $w$, and $r=1$, when the old wedge has one circle.

Each point of any of the three sets can be joined within its circle arc or circle to $w$, so [F4] makes all three path-connected. At the basepoint $w$, step 3.1 and [F5] identify the overlap fundamental group with that of a point, hence with the one-element group. For any other basepoint $y$, a path $\rho$ from $w$ to $y$ gives an isomorphism from the group at $w$ to the group at $y$ by $[\alpha ]\mapsto [\bar{\rho }\ast \alpha \ast \rho ]$, with reverse-path inverse, using [F7]; hence its fundamental group is one-element as well. The overlap is nonempty, so [F6] makes it simply connected.