A curvilinear triangulation gives a finite regular CW complex

For each $v\in V$ take the singleton $0$-cell $\{v\}$; for each $e\in E$ take its relative interior as a $1$-cell; and for each $f\in F$ take ${\varphi }_f(({\mathrm{\Delta }}^2{)}^{\circ })$ as a $2$-cell. The compact edge interval has closed image in the Hausdorff surface by [F1] and [F6], and its relative interior is dense in that image, so its cell closure is the interval with its two endpoint vertices. By [F2], each face image is the closure of its open face, and its boundary is the three full edge cells and their vertices. The face-edge incidences and vertex-link clauses in [F3], together with the face intersection and edge intersection clauses, ensure that these open cells are pairwise disjoint and cover $M$ and that every cell boundary is a union of lower cells.

Let $X^0=V$ and $X^1=V\cup {\bigcup }_{e\in E}e$. The finite subspace $V$ is discrete: since $M$ is Hausdorff, each singleton is closed, and for each $v\in V$ the intersection of the finitely many open sets $M\setminus \{w\}$, $w\in V\setminus \{v\}$, isolates $v$ in $V$. Attach one closed interval for each edge to its two endpoint vertices using its supplied embedding from [F1]. The resulting finite attachment quotient is compact and maps continuously and bijectively to $X^1$: injectivity follows because distinct edge images meet only at common endpoints. Since $X^1$ is a subspace of the Hausdorff space $M$ by [F6], the map is a homeomorphism; a continuous bijection from a compact space to a Hausdorff space is closed because the image of every closed subset is compact and hence closed. Now attach one triangular disk for each face along its boundary map from [F2]. For completeness, let $c$ be the triangle's centroid; each ray from $c$ meets the polygonal boundary once, at a positive continuous radial distance $R(u)$ for unit direction $u$, so the map $c+rR(u)u\mapsto ru$ for $0<r\le 1$, extended by $c\mapsto 0$, is a homeomorphism from the closed triangle to the closed unit disk (continuity at $c$ follows as the image norm is $r$). Thus the supplied face map gives a disk attachment as in [F4]. The face quotient is compact and maps continuously and bijectively onto $M$: [F2] gives injectivity on each face, while [F3] and step 1.1 show distinct face interiors are disjoint from each other and from $X^1$, with all remaining identifications exactly along their common full edges or vertices already in $X^1$. The compact-to-Hausdorff argument using [F6] makes this map a homeomorphism. The open interval and face interiors map homeomorphically onto the cells of step 1.1; taking $X^n=M$ for $n\ge 2$ completes the filtration required in [F5].

There are finitely many cells, so every closed cell meets only finitely many cells. Each face image is closed by [F2]; each edge image is compact by [F1] and hence closed in the Hausdorff space $M$ by [F6]; each vertex is closed as well. These finitely many closed cells cover $M$. If $A\subseteq M$ has closed intersection with every closed cell, each such intersection is closed in $M$ because the closed cell is closed in $M$, and their finite union is $A$; hence $A$ is closed. The converse follows by restriction to each closed cell. This is the weak topology required in [F5].

The vertex characteristic map is the point inclusion, each edge characteristic map is its supplied interval embedding, and each face characteristic map is the composition of the fixed disk-to-triangle homeomorphism in step 2.1 with its supplied face homeomorphism. Thus every characteristic map is a homeomorphism onto its closed cell, which is the regularity asserted in the statement. The open cells are indexed once each by $V$, $E$, and $F$, giving the stated counts. If $M$ is empty, the covering clause forces all three finite index sets to be empty and the empty CW complex satisfies the same clauses. The supplied finite data require no choice principle.