Whether the $FS$-domain of closed discs in the plane is a retract of a bifinite domain is a long-standing open problem in domain theory, which has been stated in textbooks and several other references. In this paper, we give a positive answer to this question. As preparation, we present some rules for the inductive construction of a Plotkin-poset. Then, we inductively construct a bottomed Plotkin-poset based on closed circular rings in the plane with rational radii and rational centers. Since every element in this bottomed Plotkin-poset is compact, it follows that the ideal completion of this Plotkin-poset is a bifinite domain. Furthermore, an embedding-retraction pair is established between the domain of closed discs in the plane and this bifinite domain.

We construct a model for weak $(\infty ,0)$-categories based on the theory of opetopes. The combinatorial complexity of opetopes, even just positive ones, presents a significant obstacle to developing model structures on the corresponding presheaf category. We overcome this by defining a full subcategory $\widehat {\textsf{pOpe}_{\iota}}_{EZ}$ of $\widehat{\textsf{pOpe}_\iota}$, which we prove to be a reflective subcategory of presheaves on $\textsf{pOpe}_\iota$ and possesses the regularity needed for homotopical arguments.

Our main results are the following:

• • The category $\widehat{\textsf{pOpe}_\iota}_{EZ}$ carries a cofibrantly generated model structure for $(\infty ,0)$-categories, constructed via a modification of the Cisinski–Olschok theory.

• • This model structure is Quillen equivalent to the Kan–Quillen model structure on $\textsf{sSet}$.

These results establish $\widehat{\textsf{pOpe}_\iota}_{EZ}$ as a valid opetopic model for spaces, while the underlying combinatorial framework provides a foundation for future development of opetopic weak $(\infty ,1)$-model.