Finite triangulation of a compact connected surface

Assume AC and choose, for every $p\in M$, a disk chart ${\theta }_p:B(0,2)\to U_p$ with ${\theta }_p(0)=p$. Put $Q_1=[-1\mathrm{/}2,1\mathrm{/}2{]}^2$ and initially $Q_2=[-1,1{]}^2\subset B(0,2)$. The sets ${\theta }_p(\mathrm{int}{Q}_1)$ cover $M$; compactness gives finitely many charts indexed $1,\dots ,n$ whose $Q_1$ interiors still cover. We allow the chart domain disks to be represented by disjoint formal copies; their images in $M$ overlap. The later boundary changes preserve the containment of every original ${\theta }_i(Q_1)$ in the interior of its new closed chart disk.

Replace the initial Q2 by a closed axis-aligned rectangle still contained in D_k and containing Q1 in its interior. Choose its four side levels independently in small allowed intervals near the initial side levels. For each embedded circle, its maximal nondegenerate horizontal or vertical straight subarcs are countable: their relative interiors pull back to pairwise disjoint nonempty open parameter intervals on S¹, and each interval contains a distinct rational parameter after fixing a circular coordinate. Exclude the countably many corresponding horizontal and vertical line levels and the finitely many coordinate levels of old-old crossing points. The four selected sides then contain no old-curve subarc, and no corner or side contains an old-old crossing point. This removes shared C=∂Q2 subarcs; isolated or Cantor contacts are allowed. Choose Q1⊂int Q_n⊂Q_n⊂int Q_w⊂Q_w⊂int Q2, with Q_n and Q_w closed polygonal squares. For an older circle C_j, consider each component I of the parameter preimage of θ_k(int Q2) along its parametrization γ_j:S¹→M. This uses θ_k^{-1} only on actual points in U_k. Call I relevant if its image meets Q_w. Distinct proper components I are disjoint open parameter intervals with endpoints mapping to θ_k(C). Put A=γ_j^{-1}(θ_k(Q_w)) and B=γ_j^{-1}(θ_k(C)), compact disjoint subsets of the parameter circle S¹. Their positive parameter-metric separation is η>0. Every relevant interval contains a point of A and has endpoints in B, so its circular interval length is at least η. There can be only finitely many pairwise disjoint intervals of that length on S¹. This argument uses only the global old-circle parametrization γ_j:S¹→M and compact preimages, not an undefined global chart pullback. An entire S¹ component is one wholly interior Jordan circle, also finite. For a proper I, its closure in S¹ maps injectively to a simple C-to-C arc with distinct endpoints. If its two endpoint images were the same, injectivity forces the endpoints to be the same point of S¹; then I=S¹{t} and its closure is the whole old Jordan circle tangent to C at one point. Include that as a circle piece J, as well as each wholly interior circle. Thus the finite relevant family consists only of distinct-endpoint C-to-C arcs and circle pieces J, with every J contained in cl(int C) and meeting C in at most one point. Pairwise intersections are finite: old-old intersections were finite by the induction invariant; two components of one old circle have disjoint interiors and can meet only at their finitely many listed C endpoints. No relevant piece shares a C subarc by the generic side choice. All nonrelevant components are disjoint from Q_w. Later choose the new polygon Q'' strictly within Φ(int Q_w), around Φ(Q_n); then Φ^{-1}(Q'') cannot meet any nonrelevant component, regardless of infinitely many outer contacts. The wider/narrower separation is essential.

Start with polygonal C and all finitely many relevant C-to-C arcs with distinct endpoints. Subdivide at their finitely many mutual intersections and all C contacts, adding degree-two vertices if needed to make a finite simple graph. It is 2-connected. To see this, delete any vertex v. C-v remains connected. Each original C-to-C arc minus v has at most two components, each containing one of its two distinct C endpoints unless that endpoint is v; in that case the other component/remaining arc reaches its other C endpoint. Every surviving point of every arc remains attached to C-v. Hence the full graph minus v is connected. Additional subdivisions preserve this property. The outer face is still the exterior of $C$.

Let H be the currently included finite 2-connected graph containing C, drawn by arbitrary simple arcs with finite intersections. All its bounded faces are Jordan regions by [F1]. Let K be the finite union of all remaining relevant circle pieces. The full arrangement H∪K is compact, and its finite vertex set V consists of H vertices plus all H-K and K-K crossings/contacts. Assume no shared subarcs, by the generic side choice of step 2.1. If K is nonempty, choose a circle J0 with a free point x in the relative interior of a topological edge of J0\H; if none exists, the remaining circles are already fully represented in H. If J0 meets H in at least two distinct points, it may be added immediately: a cycle meeting a 2-connected graph in two distinct vertices preserves 2-connectivity by the vertex-deletion test. Otherwise J0\H lies in one bounded Jordan face F of H, and choose x in that face away from V. Here “free” is topological; no differentiability of an old Jordan arc is assumed. Choose a relatively open facial-edge subarc $A$ of $\mathrm{\partial }F$ disjoint from $K$ and from $V$; finite contact and absence of shared subarcs give one. Choose a compact nondegenerate subarc $A_0\subset A$ and a point $a$ in its relative interior. As $K$ is compact and disjoint from $A_0$, the distance $\delta =\mathrm{dist}(A_0,K)$ is positive. Choose an interior point $z$ of $F$ close enough to $a$ that every nearest point of $z$ on $\mathrm{\partial }F$ lies in $A_0$ and $2\mathrm{\mid }z-a\mathrm{\mid }<\delta$; this is possible by taking $z$ on the $F$ side near $a$. Let $y$ be one such nearest point. The open ball $B(z,\mathrm{\mid }z-y\mathrm{\mid })$ lies in $F$ and misses $K$, since $\mathrm{\mid }z-y\mathrm{\mid }\le \mathrm{\mid }z-a\mathrm{\mid }$ and $\mathrm{dist}(z,K)\ge \delta -\mathrm{\mid }z-a\mathrm{\mid }>\mathrm{\mid }z-a\mathrm{\mid }$. Thus $(z,y]$ reaches $y$ from one component of $F\setminus K$ with no other boundary contact. The point $y$ is strongly accessible without any assumed smoothness of the wild edge. The open connected Jordan face $F$ minus the finite set $V$ is polygonally connected, so take a polygonal path in $F\setminus V$ from $z$ to $x$, made transverse to the finitely many polygonal segments already present. Stop it at its first contact $p$ with compact $K$. Because the path misses $V$, $p$ is a free point of a single remaining Jordan circle $J$; its preceding straight segment lies in a component $U$ of $F\setminus K$ and reaches $p$ from $U$. The initial path from $z$ to $p$ can be loop-erased, joined to the straight access segment $y$ to $z$, and shortened at any repeated point to yield a simple polygonal connector $\alpha :y\to p$ whose interior lies in $U$. At both endpoints the connector has a straight final segment. Choose a local Schönflies chart at either free endpoint in which its incident Jordan boundary arc is a diameter; shrink the chart neighbourhood inside a Euclidean ball disjoint from every other finite graph piece, possible because those compact pieces avoid the endpoint. The connector enters the $U$ halfdisk side. These local collars are topological, while the displayed straight segments give the polygonal access actually used; the wild boundary arc need not meet an ordinary Euclidean ball in one connected interval. This finds an exposed $J$ even when other circles are nested behind it. Here $F\setminus V$ is path connected: detour a polygonal path around each of its finitely many removed points inside a small disk in $F$. The first-contact path need not avoid $K$; stopping it finds the exposed circle.

Subdivide H and J at their finite common points and at connector endpoints. If |J∩H|≥2, the union H∪J is 2-connected: J with the common points as vertices is a cycle, and deleting any vertex leaves H-v connected plus each surviving J component attached through at least one of two distinct common points. If |J∩H|=1 at v, choose the exposed connector p→y with p≠v on J and y≠v on H. Deleting v leaves J-v joined to H-v by the connector; deleting a vertex of the connector leaves J connected to H at v; all other deletions are immediate. Thus the union is 2-connected. If $\mathrm{\mid }J\cap H\mathrm{\mid }=0$, begin with the simple polygonal connector $\alpha :y\to p$ obtained above, where $y$ and $p$ lie on disjoint Jordan boundary components of the open face $U$. A sufficiently small regular neighborhood of the compact middle part of $\alpha$ is a polygonal ribbon inside $U$, with disjoint left and right long sides. To make this precise, choose disjoint endpoint disks that meet $\alpha$ only in its final straight segments, and let ${\alpha }_0$ be the closed subarc left after removing their endpoint interiors. The compact set ${\alpha }_0$ has positive distance from the closed complement of $U$; choose a finite polygonal strip about it thinner than half that distance. Shrink the endpoint disks and strip width together so the two strip ends land on opposite local sides of $\alpha$ in each endpoint disk. At $p$, choose a small Schönflies chart whose halfdisk models the $U$ side of the free Jordan subarc of $J$; choose it to avoid every other finite graph piece. The image of the initial $\alpha$ germ is still a proper embedded arc from the diameter point into that halfdisk. Stop it at its first exit from a smaller closed halfdisk. This is an arbitrary simple crosscut of a Jordan disk. The truncated crosscut has two local sides at each interior point because it is the homeomorphic image of the original straight endpoint segment. The crosscut together with each of the two diameter/boundary subarcs forms a Jordan curve. Apply the arbitrary-simple-arc crosscut proof in lem-planar-facial-graph-isomorphism-extension to the halfdisk boundary and this truncated arc: the halfdisk minus it has exactly two open connected Jordan regions, whose frontiers contain the opposite relative open diameter subarcs. Pull these regions back; their frontiers then contain the two free relative open subarcs of $J$ adjacent to $p$, and they are the left and right local $U$ sectors of $\alpha$, even though the Schönflies image of $\alpha$ need not be straight. The same construction at $y$ uses the free facial subarc of $H$. The strip ends just constructed lie in the corresponding local sectors at $p$ and $y$; label those sectors by the two sides of the strip, reversing a local plus/minus name at $y$ if necessary. In each of these four open sectors select a boundary point $q$ in its free $J$ or $H$ subarc away from $\alpha$ and the other finite graph pieces. Pick an interior point $z$ close enough to $q$ that its nearest point on the sector frontier lies in a smaller relative subarc about $q$, which has positive separation from every other local frontier piece. The nearest-ball argument of step 4.1 gives a straight segment from $z$ to a distinct free point $p_{+},p_{-}$ or $y_{+},y_{-}$; its relative interior stays in that sector. Polygonal connectivity of the corresponding sector connects its interior end to the matching long side of the middle ribbon. These four short connecting pieces can be chosen in disjoint endpoint disks and on opposite sides of $\alpha$; erase any self-loops within each side. The result is two disjoint simple polygonal connectors in $U$, $p_{+}\to y_{+}$ and $p_{-}\to y_{-}$, with four distinct free endpoints. This is the required two-attachment lemma for arbitrary Jordan arcs: the Euclidean ribbon is used only away from wild endpoints, and the endpoint access is supplied by nearest balls in the two local topological sectors. Deleting any vertex leaves one attachment or the other, so $H$ together with $J$ and these two connectors is 2-connected. In all cases subdivide circle arcs and parallel connector arcs as needed to keep the finite abstract graph simple and at least three vertices. Each step adds one previously missing circle piece and uses connectors whose interiors lie in a face of the full remaining arrangement, so connectors meet no unprocessed wild circle. The number of remaining circle pieces decreases by one; the process terminates after finitely many steps. Every later circle meets the already included graph only at the originally finite old-old/old-C contacts. The outer C stays a fixed graph cycle and all additions lie inside it.

Use the rooted ear decomposition of the final 2-connected finite graph from the fixed cycle C (finite ear lemma Step1.2). Begin with the same polygonal C in the target plane. Maintain an ambient homeomorphism h_i carrying the source partial graph to a polygonal partial graph, fixing C pointwise, and a face bijection induced by h_i. For the next source ear P, its relative interior is disjoint from the partial graph and connected, hence lies in one bounded Jordan face F (all additions are inside C). Its distinct endpoints lie on the Jordan frontier of F. The target arc h_i(P) lies in the paired polygonal Jordan face F'. Choose short straight endpoint germs in the corresponding F' sectors, then a simple polygonal path through connected F' joining them and avoiding the existing target graph; mark along it, in order, as many distinct interior points as $P$ has internal graph vertices, and call the resulting polygonal ear $P^{\mathrm{\prime }}$. The Jordan crosscut lemma splits F' along each of h_i(P) and P' into two Jordan disk closures whose boundaries are the corresponding subpath of ∂F' together with the crosscut. Fix ∂F' pointwise and choose a homeomorphism $h_i(P)\to P^{\mathrm{\prime }}$ fixing endpoints and taking each internal ear vertex to its corresponding ordered mark. The resulting prescribed boundary homeomorphism on each pair of split disks extends over that pair by Jordan–Schönflies. The two extensions agree along h_i(P), and both restrict to identity on ∂F'; paste them to a homeomorphism g_i of cl F' and extend g_i by identity outside F'. Then h_{i+1}=g_i∘h_i is an ambient plane homeomorphism fixing C, carrying the new ear to P', and pairing the two new faces by their labelled boundary subpaths. All unchanged faces keep their old pairing. The finite induction produces Φ fixing C and a full β pairing facial cycles and the outer face. It directly supplies the premises of the graph-extension theorem; no unsupported inference from Gallier–Xu Proposition E.3 to β is needed. Gallier–Xu Appendix E Proposition E.3 (PDF p170 / printed p160) supports rooted polygonal ear redraw only. Its Theorem E.2 on the same page explicitly requires facial and outer-face matching. The preceding induction supplies those requirements after finite augmentation.

Apply Φ to Q_n,Q_w,Q1 and all older local arcs. The relevant graph is polygonal. Choose a simple polygonal Jordan boundary ∂Q'' between Φ(Q_n) and Φ(∂Q_w), enclosing Φ(Q1), by the following nested polygonal separator lemma; Q'' denotes the closed disk it bounds. For compact nested Jordan disks K⊂int L, apply [F5] with compact K and open U=int L. It supplies a finite polygonal chain Γ with trace in U\K and index n(Γ,x)=1 for every x∈K. Subdivide the finitely many polygonal chain segments at every segment crossing and at every endpoint of a collinear overlap, then combine coincident subsegments and orient negative coefficients backward. The resulting finite directed multigraph is balanced at every vertex because the chain boundary is zero. Starting with any directed edge, follow outgoing edges until a vertex repeats, remove that directed circuit, and continue; balanced degree ensures this exhausts all edges. Split each circuit at repeated vertices. Thus Γ is a finite sum of oriented simple polygonal cycles γ_i. By additivity of index ([F6]), at a fixed x∈K the integers n(γ_i,x) sum to 1, so one is nonzero. Its trace is disjoint from connected K, and index is constant on K by [F6]. The Jordan curve theorem gives a unique bounded complementary region; index vanishes in the unbounded region, so nonzero index places all K in that bounded region. Since γ_i⊂int L and the exterior of the Jordan disk L is unbounded and connected, the closed disk bounded by γ_i lies in int L. This γ_i is the required polygonal separator. Perturb its finitely many vertices by less than half the positive distances of its trace from $K$, from $\mathrm{\partial }L$, and between its finitely many nonincident closed edges, and choose the perturbation generically so it crosses polygonal relevant edges only finitely and transversely, away from graph vertices and old-old crossings, with no triple contacts. The straight-line homotopy from the original polygonal cycle to its perturbed version avoids every point of $K$ and stays inside $L$, so winding index at each point of $K$ is unchanged; hence the perturbed Jordan cycle still encloses $K$. Every nonrelevant old component lies outside Φ(Q_w), so Q'' misses it. Set θ'_k=θ_k∘Φ^{-1} on Φ(D_k), and use Q'' as the new closed chart disk. Its image contains the old θ_k(Q1), so the finite interior cover persists. The new boundary is Jordan and meets each earlier boundary in finitely many topologically transverse points with no triples. The induction invariant refers only to processed pairs $i<j\le k$; a pair is made finite at its younger index. After $k=n$ all pairs are finite and topologically transverse, with no triple contact.

Let Γ=∪_i ∂Θ_i. Make every pairwise crossing a vertex and choose at least three extra vertices on every circle if needed. Between these finitely many marks, the circles give finitely many embedded edges; distinct circles have no shared subarc because all their contacts are isolated transverse points. Γ is a finite graph embedded in M. Every component R of M\Γ has nonempty frontier in Γ: otherwise it would be open and closed in connected M. Every frontier point has a local disk in which Γ consists of one embedded arc or two transverse arcs, so each local sector is adjacent to an open edge side. In particular R touches a side of some edge. A side germ of an open edge belongs to one complement component, and there are finitely many edge sides, hence finitely many R. For each i, R is disjoint from ∂Θ_i. The sets R∩int Θ_i and R\Θ_i are relatively open, disjoint, and cover R, so connectedness puts R entirely inside int Θ_i or entirely outside Θ_i. The interiors cover M, so select i(R) with R⊂int Θ_i(R). This is one choice for each member of a finite set. The closure of R may meet ∂Θ_i(R), which is harmless: the local graph includes that circle as its outer cycle.

Fix R and i=i(R). Pull Γ∩Θ_i into the closed plane Jordan chart disk; after a Schönflies boundary homeomorphism it may be viewed inside a closed square C. Each other boundary circle intersects C in finitely many transverse points, with no shared subarc. Its part in the square is a finite union of proper C-to-C arcs with distinct endpoints, or a whole Jordan circle strictly inside C; a circle disjoint from C and outside contributes nothing. Pairwise intersections of these finitely many pieces remain finite. Thus this local graph satisfies the finite arrangement premises of steps 2.1–5.1, with no same-endpoint tangent case needed after transversality. Add C and all C-to-C arcs to get a 2-connected root graph. Process remaining whole circles by the full-arrangement exposed-face construction: 0 common vertices require two disjoint face connectors, 1 requires one, and ≥2 require none. Connector interiors avoid all original local curve pieces, including unprocessed circles. The number of circles decreases each step, so the resulting H_i is a finite 2-connected simple plane graph after harmless finite subdivisions. Every added connector lies in a single original component of int Θ_i\Γ, since its relative interior avoids the whole original Γ. By [F1], every bounded face of H_i is a Jordan region whose closed frontier is an embedded graph cycle, and every edge has a face on each side. H_i includes the entire local Γ∩Θ_i. Hence H_i's faces inside the chosen original region R are Jordan regions, and the added connector arcs with relative interior in R partition R into finitely many of them. Retain only those added arcs lying in R, together with their endpoints; discard the additions in other original regions. For different R, repeat independently. Since distinct original regions are disjoint, the retained arc interiors cannot meet across different choices, and all endpoints lie on original Γ or on other retained arcs from that same R. Make the finitely many endpoints common global vertices, choosing distinct free-edge endpoints when possible. The final global Γ' is a finite embedded graph containing Γ. Its complementary regions are exactly the retained local H_i faces, so every region F has a closed Jordan disk closure embedded in M with boundary a simple graph cycle. The two-attachment case is needed here: a single slit of an annulus repeats a boundary shore, while the augmented graph has embedded Jordan facial frontiers.

Subdivide every edge of Γ' finitely until it has distinct endpoints and every Jordan facial cycle has at least three distinct edges and three distinct vertices. Edge subdivision does not alter faces or 2-connectivity of the local witness graphs. The vertices, open edge interiors, and open Jordan faces now form a finite regular CW structure on M: each edge closure is an embedded closed interval, each face closure is an embedded closed disk, and distinct open cells are disjoint. Every graph edge has two distinct final adjacent faces. A retained connector lies in one original region $R$ and is an edge of its local $H_i$; the two $H_i$ faces on its sides lie in $R$ and cannot reconnect outside the chart because $R\subset \mathrm{int}{\mathrm{\Theta }}_i$. For an original $\mathrm{\Gamma }$ edge, side germs in different original regions remain different; if both lie in one $R$, that edge interior lies in $\mathrm{int}{\mathrm{\Theta }}_i$ and its two local $H_i$ faces in $R$ remain distinct. Thus no bridge or edge bordering the same final face twice remains, and each Jordan facial boundary uses an edge at most once. The local half-plane and transverse-crossing sector charts show each vertex has a single cyclic sequence of incident face corners and edge germs. Fix one homeomorphism ${\gamma }_e:[0,1]\to \bar{e}$ for each finite global open graph edge $e$, taking the parameter endpoints to its two graph vertices. For each face F take a convex polygon P_F with one side per boundary-edge occurrence, in cyclic order. Define its boundary homeomorphism to the graph cycle ∂F side by side: on each straight polygon side use the affine side coordinate $t\in [0,1]$ followed by the fixed ${\gamma }_e$, reversing $t$ when that face traverses $e$ backwards. These maps agree at polygon corners because ∂F is a simple graph cycle, so they assemble into one boundary homeomorphism; extend it over the closed disk by Jordan–Schönflies. Pair corresponding polygon sides by those already fixed boundary parametrizations. Because both copies use the same ${\gamma }_e$, each paired-side identification is exactly the affine map $t\mapsto t$ or $t\mapsto 1-t$ in the side coordinates, despite the wild embedding of $e$ in $M$. The resulting finite polygon quotient maps continuously and bijectively onto M: interiors map to distinct F, open sides to the single corresponding graph edge, and corners to the graph vertices. Compact-to-Hausdorff makes it a homeomorphism. Because each face has at least three distinct boundary vertices, one may triangulate each convex P_F by a fan from an interior point (one triangle per boundary edge). The quotient is a finite regular triangular cellulation; the center is unique to its face and each triangle has three distinct vertices. A boundary edge has exactly two incident triangles, one from each adjacent face; a radial edge within P_F has exactly two incident fan triangles.

Let P be the full incidence poset of vertex, edge, and triangle cells of that regular triangular cellulation, ordered by inclusion of closures. Include all proper incidences, in particular vertex<triangle even when the intervening edge is omitted from a two-element chain. Let K=Δ(P), whose vertices are cells and whose 2-simplices are strict chains vertex<edge<triangle. Within each closed triangle, these chains are exactly the six barycentric subtriangles. Regularity makes the piecewise affine barycentric maps agree on common subedges and gives a continuous bijection |K|→M; compact-to-Hausdorff makes it a homeomorphism. Every edge of K is one of three types: vertex<edge, edge<triangle, or vertex<triangle. The first lies in exactly two 2-simplices because each original edge has two adjacent triangles. The second lies in exactly two because an original edge has two distinct endpoints. The third lies in exactly two because each original triangle has exactly two edges incident to that vertex. Each vertex link of K is a finite connected 1-manifold: its triangles around a point follow the single cyclic sector order of M, and every link vertex has degree two by the preceding edge count; therefore each link is one cycle. K is finite and 2-dimensional, as required.

The complex $K$ of step 11.1 is finite and two-dimensional, its realization is homeomorphic to $M$, each of its edges lies in exactly two triangles, and each vertex link is a cycle. The only unbounded choice is the chart selection in step 1.1; all later selections are finite, while the Jordan and graph suppliers inherit their stated AC hypotheses. This proves the statement.