Subanalytic triangulations on analytic manifolds

This reading constructs compatible locally finite triangulations of arbitrary Hausdorff second-countable real analytic manifolds. It supplies the relative Euclidean construction, elementary finite-colour chart covers, differentiable subanalytic cutoffs, a proper embedding and analytic recovery of the transported open simplices. Six exercises have complete solutions.

Original programme text and solutions by GPT-6.1 Sol (OpenAI), with mathematical contributions by GPT-6 Astra (OpenAI). Original programme text: public domain (CC0).

Setting and dependency boundary

Manifolds have finite dimension and no boundary. The prescribed family is locally finite, and each member is locally subanalytic. The compatible analytic partitions and simultaneous constant-rank refinements are proved below from the linked finite-cell calculus and compact chart covers. The relative construction uses the linked compact uniformization theorem and the exact analytic coordinate and uniqueness results compared below. Real-analytic Noetherianity is proved below; we also use the analytic constant-rank theorem and differential-equation facts, including uniform uniqueness. The dimension and frontier results have their complete linked provider below. The uniformization application is written out in the singular-line proof below; the analytic input comparison states the remaining coordinate and uniqueness uses precisely. The finite-colour cover and cutoff proofs do not assume a triangulation or a covering-dimension theorem.

The bounded-chart comparison and global Boolean/projection calculus used for proper images have their complete written provider in analytic finiteness and preparation. Guillaume Valette is credited there for these results.

Real-analytic Noetherianity

Proposition (real analytic germs). For every integer $n\geq0$, the ring $A_n=\mathbb R\{x_1,\ldots,x_n\}$ of convergent real power series is Noetherian: every ideal has a finite generating set. Consequently, the local ring of real-valued real analytic germs at any point of a finite-dimensional real analytic manifold is Noetherian. Every ascending sequence of ideals in such a local ring stabilizes.

Proof. The complex-coefficient statement is Theorem 6 of Weierstrass preparation and division. Its proof uses division by a regular germ (Corollary 3), a linear change making a nonzero germ regular (Proposition 4), and induction on the number of variables. It does not use a triangulation. We explain the change of coefficient field, since a subring of a Noetherian ring need not itself be Noetherian.

Put $B_n=\mathbb C\{z_1,\ldots,z_n\}$. The coefficient-preserving map $\iota:A_n\longrightarrow B_n$, sending $\sum_\alpha a_\alpha x^\alpha$ to $\sum_\alpha a_\alpha z^\alpha$, is an injective ring homomorphism. Indeed, convergence here means absolute convergence on some sufficiently small polydisc, a condition on $\sum_\alpha |a_\alpha|r^\alpha$ that is the same for real and complex variables. Uniqueness of the Taylor coefficients gives injectivity. We identify $A_n$ with this subring of $B_n$.

For $b=\sum_\alpha c_\alpha z^\alpha\in B_n$, split each coefficient into its real and imaginary parts. The two resulting series converge on every smaller polydisc on which the original series converges absolutely, because $|\operatorname{Re}c_\alpha|,|\operatorname{Im}c_\alpha|\leq|c_\alpha|$. Thus $b=u+iv$ for unique $u,v\in A_n$. This proves $B_n=A_n\oplus iA_n$ as $A_n$-modules. In particular, the coefficient projection $\rho:B_n\longrightarrow A_n$, $\rho(u+iv)=u$, satisfies $\rho(a b)=a\rho(b)$ for $a\in A_n$ and $\rho(a)=a$. It is an $A_n$-linear map; we do not claim that it is a ring homomorphism.

Let $I\subset A_n$ be an ideal. The extended ideal $IB_n$ consists of finite sums of products $bf$, with $b\in B_n$ and $f\in I$. Since $B_n$ is Noetherian, choose finitely many generators $g_1,\ldots,g_t$ of $IB_n$. Write each $g_k$ as such a finite sum, and collect all the elements of $I$ that occur into a finite list $f_1,\ldots,f_s$. The ideal generated by this list in $B_n$ contains every $g_k$ and is contained in $IB_n$; hence it equals $IB_n$. For any $f\in I$, there are $b_1,\ldots,b_s\in B_n$ with

$$f=\sum_{j=1}^s b_jf_j.$$

Applying the $A_n$-linear map $\rho$ gives $f=\sum_{j=1}^s\rho(b_j)f_j$, now with coefficients in $A_n$. Therefore $I=(f_1,\ldots,f_s)$ in $A_n$. For $I=0$, the empty list suffices. This also covers $n=0$, when the two rings are $\mathbb R$ and $\mathbb C$.

If $p$ lies in an $n$-dimensional real analytic manifold $M$, choose an analytic chart $\phi:U\longrightarrow V\subset\mathbb R^n$ with $\phi(p)=0$. Composition defines a ring isomorphism $\mathcal O^\omega_{M,p}\longrightarrow A_n$, $[h]\longmapsto[h\circ\phi^{-1}]$; its inverse sends $[a]$ to $[a\circ\phi]$. Both maps are well-defined on germs because analytic compositions are analytic and restrictions agree on smaller neighbourhoods. This transfers finite generation to $\mathcal O^\omega_{M,p}$.

Finally, for an ascending sequence $I_1\subset I_2\subset\cdots$, its union $I$ is an ideal: any two elements lie in one common stage, and each stage is closed under multiplication by the ambient ring. Choose finitely many generators of $I$. They all belong to some $I_N$, so $I=I_N$, and $I_m=I_N$ for every $m\geq N$. $\square$

This is a statement about germs at each fixed point. It does not supply one neighbourhood or generating family valid simultaneously at all points, and it does not prove subanalytic uniformization or the analytic regularity results used elsewhere in the construction.

Compatible locally finite analytic partitions

The finite analytic cell theorem, its Boolean and projection calculus, and its bounded-chart comparison give the following manifold partition. The compact chart-ball refinement is proved below in Section 2 of the cutoff construction. These are the exact inputs; no analytic regular-locus theorem or uniformization theorem is needed for this partition.

Partition theorem. Let $M$ be a finite-dimensional Hausdorff second-countable real analytic manifold without boundary, and let $\{A_a\}_{a\in I}$ be a locally finite family of locally subanalytic subsets. There is a countable locally finite partition of $M$ into connected locally subanalytic embedded analytic submanifolds compatible with every $A_a$: each partition member is contained in or disjoint from each $A_a$. Given finitely many analytic maps $f_j:M\to N_j$ to real analytic manifolds, the partition can also make every restriction $f_j|_S$ have constant rank.

Compact chart pieces and compatibility

The compact exhaustion and shell construction in the linked cutoff proof supplies a countable cover by open coordinate balls $B_i$ with compact closures $L_i=\overline{B_i}$ contained in analytic charts; the family of these closures is locally finite. For instance, use the inner balls in that construction: their closures are contained in its locally finite outer compact balls. Each ball and its closure is locally semianalytic. In its chart it has a squared-norm inequality, and outside its compact closure it is empty on a neighborhood. The exhaustion sets need not be subanalytic; only the selected balls enter the partition.

For the map assertion apply this refinement to the open cover consisting of intersections of a source chart and inverse images of target charts for the finitely many $f_j$. Thus each $L_i$ lies in one source chart, and every $f_j(L_i)$ lies in one selected target chart. The source and target coordinate images of these compact sets are bounded. The maps in these coordinates are analytic on a neighborhood of $L_i$.

Every compact set meets only finitely many members of a locally finite family. Indeed, cover the compact set by finitely many neighborhoods, each meeting finitely many family members; combine their finite lists. In particular $L_i$ meets only finitely many $L_k$ and finitely many nonempty $A_a$.

Enumerate the balls and put

$$E_i=B_i\setminus\bigcup_{k<i}B_k. \qquad\text{(P1)}$$

These sets are disjoint and cover $M$, because each point has a first containing ball. For fixed $i$ the earlier union is finite; its trace near $L_i$ uses only the finitely many earlier closures that meet a sufficiently small neighborhood of $L_i$. To justify that neighborhood, first take a finite cover of $L_i$ by neighborhoods meeting only finitely many closures, then discard the finitely many closures missing $L_i$ by shrinking around that compact set. Thus $E_i$ is a finite Boolean combination of locally semianalytic balls near $L_i$. The local Boolean calculus also makes every relevant $A_a\cap E_i$ locally subanalytic there.

In the chosen source coordinates, these traces are bounded and locally subanalytic at every point of their ambient closures. Their closures lie inside the compact coordinate image of $L_i$, strictly inside the source chart. Extension by the empty set therefore creates no chart-boundary obstruction. The proved bounded-chart comparison makes $E_i$ and all the finitely many relevant $A_a\cap E_i$ globally subanalytic in this Euclidean space.

Apply the finite analytic cell theorem compatibly with these sets, and keep just the cells in $E_i$. Each kept cell is a connected embedded analytic submanifold. Its global subanalyticity gives local subanalyticity in the source chart, including at points of its closure; compact containment gives local emptiness outside that chart. It is therefore locally subanalytic in $M$. It lies in or misses every $A_a$, including those whose trace on $L_i$ is empty.

There are finitely many kept cells for each $i$, and each is contained in $L_i$. A neighborhood meeting only finitely many $L_i$ consequently meets only finitely many kept cells. They give a countable locally finite partition of $M$. This proves the first assertion. It uses local finiteness of the compact closures; a merely countable atlas does not supply that conclusion.

Analytic coordinates on a cell

We spell out the cell coordinates needed for rank refinement. A cylindrical analytic cell $C$ of dimension $d$ has a coordinate projection $q:C\to V$ which is a definable analytic diffeomorphism onto an open definable $V\subset\mathbb R^d$. Keep precisely its band coordinates and discard its graph coordinates.

This follows by induction through the cylindrical construction. For a graph over a cell with inverse parametrization $\phi$, use $u\mapsto(\phi(u),\zeta(\phi(u)))$; the old free-coordinate domain remains open. For a band over that cell, the new free domain is

$$\{(u,t):u\in V,\ \zeta_-(\phi(u))<t<\zeta_+(\phi(u))\}. \qquad\text{(P2)}$$

It is open because the endpoints are continuous, omitting the corresponding inequality for an infinite endpoint. Its inverse parametrization is $(u,t)\mapsto(\phi(u),t)$. These maps and their inverses are analytic. Their graphs are definable by the graph, composition and projection calculus; hence the domains are definable too. In dimension zero use the single point $\mathbb R^0$. This proves the coordinate assertion, including the analytic embedding and its inverse, directly from the cell definition.

Definable derivatives and restriction ranks

Fix a kept cell $C\subset E_i$, and write $\phi:V\to C$ for this inverse parametrization. In the chosen target coordinates the maps $g_j=f_j\circ\phi$ are analytic and definable. To check the latter assertion, the coordinate graph of $f_j$ over $L_i$ is bounded and locally semianalytic at every point of its compact closure: its defining analytic equation is valid on a neighborhood of $L_i$, and the closed coordinate ball supplies the domain condition. The bounded-chart comparison gives a globally subanalytic graph. Restriction to $C$ and composition with $\phi$ preserve definability.

For a scalar analytic definable $g:V\to\mathbb R$, the derivative graph, with $u\in V$, is described by

$$\begin{aligned} v=\partial_\ell g(u) \quad\Longleftrightarrow\quad &\forall\epsilon>0\ \exists\delta>0\ \forall h\in\mathbb R,\\ &\bigl(0<|h|<\delta\ \text{and }u+he_\ell\in V\bigr)\\ &\hspace{1em}\Longrightarrow \left|\frac{g(u+he_\ell)-g(u)}h-v\right|<\epsilon. \end{aligned} \qquad\text{(P3)}$$

Function values here can be written using extra real variables in the graph of $g$; division by $h\ne0$ is a polynomial graph condition. The formula uses finitely many real quantifiers. Existential quantification is projection, and universal quantification is complement of an existentially quantified complement. Thus the established Boolean and projection calculus makes this derivative graph definable. Openness of $V$ supplies a nontrivial interval of allowed $h$ at every $u$, so the limit condition specifies the derivative uniquely. Applying this to every scalar coordinate gives definable differential matrices for all $g_j$.

All rank sets are consequently definable: rank $r$ means that every $(r+1)$-minor vanishes and, when $r>0$, some $r$-minor is nonzero. Rank zero means every matrix entry vanishes. Include all finitely many rank sets for the finitely many maps in a finite analytic cell decomposition of $V$.

Every resulting $d$-dimensional cell $D\subset V$ is open in $V$. Indeed, a full-dimensional cylindrical cell has only band steps, and (P2) proves openness at every such step. Therefore $\phi(D)$ is open in $C$; its tangent space is that of $C$, and every $f_j|_{\phi(D)}$ has the constant rank selected on $D$. Retain these members as finished.

On a smaller cell $D$, the old differential rank is insufficient: restricting a rank-one map to one of its level curves can give rank zero. Take the cell’s own free-coordinate parametrization $\psi:W\to D$, and repeat the rank calculation for all $g_j\circ\psi$. These are still definable analytic maps on an open definable domain. Their images under $\phi\circ\psi$ are connected embedded analytic submanifolds and definable in the original source chart. Repeat the same finite refinement and finish its full-dimensional members.

An unfinished branch now has strictly smaller dimension. Induction on $d$ terminates: dimension zero has rank zero for every map, and each positive-dimensional step finishes its full-dimensional cells and passes only finitely many smaller cells to already established lower-dimensional cases. The final refinement of each original $C$ is finite. Its members remain inside $L_i$, so refining the finitely many cells for each $i$ preserves countability, local finiteness and compatibility with all $A_a$. This proves simultaneous constant rank on every final member. $\square$

If $M$ is compact, this locally finite partition is finite by the compactness argument above. Similarly a compact subanalytic subset contained in one analytic chart has a finite compatible analytic partition, by bounded-chart comparison and finite cells. A connected cell lies in a single connected component; hence each component of a finite union of cells is a union of whole cells. There are finitely many components. For a compact subanalytic subset meeting several charts, intersect with finitely many compact coordinate boxes whose interiors cover it, apply the same finite-cell argument in each box, and take the resulting finite union of connected cells. This proves finiteness of its connected components as well.

The theorem supplies compatible analytic partitions and constant-rank refinements for the relative triangulation construction. Whitney and Verdier conditions, a frontier condition between partition members, and the intrinsic analytic regular-locus theorem require additional arguments.

Sources. Guillaume Valette, On subanalytic geometry, arXiv:2507.23622v1, §1.2, gives the graph/band cell definition and cell theorem; §2.1 gives first-order definable formulas and the definability of derivatives. The linked analytic finiteness reading proves its finite-cell and Boolean/projection inputs in the stated projective product convention. The manifold globalization, explicit derivative-limit formula, and simultaneous restriction-rank induction are written out here.

Analytic inputs and their exact uses

The analytic coordinate results used below are the inverse, implicit and constant-rank theorems, (LAF1)–(LAF4). Their proof first complexifies a convergent real power series, applies the holomorphic inverse theorem to the parameter-preserving block map, and uses uniqueness under conjugation to restrict its inverse to real points. The constant-rank proof then uses an invertible minor as part of a coordinate system; vanishing of the remaining transverse derivatives makes the other output coordinates independent of those directions. No uniformization or triangulation is an input to these coordinate theorems.

Here are the hypotheses at their uses in this construction.

1. Local equations for a partition member. A member of the proved analytic partition is an embedded analytic submanifold. Near a point of a member of dimension $d<n$, analytic submanifold coordinates give real analytic defining functions $H_1,\ldots,H_{n-d}$ with independent differentials and common zero set exactly that member in a smaller open neighborhood. The scalar function $h=\sum_j H_j^2$ has precisely the same real zero set. We do not assert that $dh$ is nonzero; the finite-jet argument uses the analytic function $h$, not a regular-value assertion about $h$.

2. A section of the direction map. Suppose the analytic map $q:W\to\mathbb {RP}^{n-1}$, $q(w)=[v(w)]$, has surjective differential at $w_0$. In source and target charts choose a nonzero maximal minor. It remains nonzero on a smaller neighborhood. There the map is a submersion, and the analytic coordinate theorem gives a section $s:D\to W$ on an open target chart, with $q\circ s=\operatorname{id}_D$. Shrink $D$ so that the sphere-to-projective-space covering has an analytic local inverse agreeing with $v(s(q(w_0)))$. By continuity, $v\circ s$ is this same inverse throughout a smaller connected chart. Thus the incidence argument really has analytic functions $x\circ s$ and a local unit direction $v$, as required for its determinant calculation. This is a local section, not a global choice of direction representatives.

3. Constant rank and finite fibres. The partition is refined using the rank of each restriction to its own tangent space. On a resulting analytic member $S$ of dimension $d$, let the projection have constant rank $r$. The coordinate normal form makes its local fibres manifolds of dimension $d-r$. If the original fibres are finite, $d-r>0$ is impossible: an open coordinate neighborhood in a positive-dimensional fibre has infinitely many points. Therefore $r=d$. This proves the local rank assertion only; the later compactness, covering and height-order arguments are still needed for the finite graph decomposition.

4. The transported open simplex. In the globalization proof, analyticity of $t|_{\sigma^\circ}$ follows from the explicit analytic chart-coordinate ratios. The equality $T|_{\sigma^\circ}=F\circ t|_{\sigma^\circ}$, with $F$ of class $C^1$, then makes $dt$ injective because $dT$ is injective. Full source rank holds at every point of this open simplex, so the analytic coordinate theorem supplies local immersion charts. The separately proved homeomorphism onto the image gives the subspace topology and makes the image an embedded analytic submanifold. Analyticity or bijectivity alone would not supply that conclusion; the example $u\mapsto u^3$ below records why the differential argument matters.

The differential-equation input is narrower than analytic dependence of a flow on all its data. The needed result is the uniform zero-solution argument for (LAF9). For clarity, suppose a local analytic relation is

$$g_{q+1}(x,v)=\sum_{j=0}^{q}a_j(x,v)g_j(x,v).$$

For initial parameters $(x,v)$, put $U_j(t)=g_j(x+tv,v)$. Differentiation gives $U'_j=U_{j+1}$ for $j<q$ and $U'_q=\sum_{j=0}^q a_j(x+tv,v)U_j$. Choose a smaller closed parameter box inside the relation’s domain and a positive time interval on which all its line segments remain there. The resulting matrix $A(t,x,v)$ has a common finite operator-norm bound $B$ on that compact product. For zero initial jets, the integral equation gives

$$\sup_{|t|\leq\epsilon}\|U(t)\| \leq B\epsilon\sup_{|t|\leq\epsilon}\|U(t)\|.$$

The supremum is finite by continuity. Taking $\epsilon$ within the chosen time interval and $B\epsilon<1$ forces it to be zero; if $B=0$, it is zero without a further restriction. The argument covers both signs of time and every parameter in the same smaller box. It proves the neighborhood assertion used in the incidence rank calculation, without asserting one interval for all charts, existence of a global flow, or analyticity of an arbitrarily given smooth solution.

The stronger intrinsic analytic regular-locus theorem recognizes every point at which a subanalytic set is locally an analytic manifold, independently of a chosen partition. That theorem is retained in its separate reading. The construction here uses the supplied analytic partition members, their constant-rank refinements and the coordinate arguments above; it does not identify their union of top-dimensional members with the intrinsic regular locus. Thus these local inputs should not be confused with that stronger recognition theorem. The finite-cell, Boolean/projection and frontier results remain the explicitly linked inputs stated above, and the graph-completion and relative-polyhedron arguments remain separate parts of the proof.

How a compatible triangulation is assembled

The triangulation input has a useful relative form: one may adapt a fixed locally finite polyhedron while preserving every old simplex setwise. The construction below follows the geometric method of Masahiro Shiota, Piecewise linearization of real analytic functions. We make its fibre ordering and straight-to-curved map explicit.

The lower subanalytic prerequisites used in this construction are compact uniformization, the set calculus, and the following explicitly identified coordinate and dimension results. A nonempty subanalytic set has lower-dimensional frontier $\overline A\setminus A$, as proved in Dimension, fibrewise closure and the frontier. This is distinct from the topological boundary, whose dimension can equal the dimension of the set. The compatible analytic partition theorem above proves that a locally finite family has a compatible locally finite partition into analytic submanifolds and that finitely many analytic maps have constant-rank restrictions after refinement. Compact subanalytic sets consequently have finite such partitions and finitely many connected components. These are inputs to the construction, rather than consequences of the triangulation being constructed. The real-analytic Noetherianity proposition above, the analytic constant-rank theorem and uniqueness for finite systems of analytic ordinary differential equations are also used.

Relative polyhedron theorem. Let $K$ be a finite-dimensional locally finite linear simplicial complex with closed support in $\mathbb R^N$. Let $\{A_i\}$ be locally finite in the ambient space, with each $A_i$ subanalytic and contained in $|K|$. There are a locally finite subdivision $K'$ and a subanalytic homeomorphism

$$T:|K|\longrightarrow |K|$$

such that $T(\sigma)=\sigma$ for every old closed simplex $\sigma$, each restriction to an open simplex of $K'$ is an analytic diffeomorphism onto a subanalytic analytic submanifold, and every such image is contained in or disjoint from each $A_i$. No analyticity across simplex boundaries is asserted.

We first prove the ingredients that make the induction possible.

Choosing a point from which lines have finite intersections

Call a line singular for a set if it contains a nontrivial interval in that set. For a compact subanalytic set with no such intervals, every intersection with a line is finite: it is a compact zero-dimensional subanalytic set. For any countable collection of subanalytic sets of dimension less than $n$, the union of their singular lines is meagre in $\mathbb R^n$. Their singular parallel directions form a meagre subset of $\mathbb{RP}^{n-1}$. Here, a meagre set is a countable union of nowhere dense sets.

Proof. A compatible analytic partition reduces the claim to countably many analytic submanifolds. A line interval in the original set contains an interval in one partition member: on a compact subinterval only finitely many members occur, and their intersections with the line are subanalytic. Work on a small open set where one such member is the zero set of an analytic function $h$; a sum of squares of local defining functions supplies $h$ in higher codimension.

Choose analytic coordinates for a unit direction $v$. Put $L=v\cdot\partial_x$ and $g_j=L^j h$. The germ of the ascending chain of ideals $(g_0),(g_0,g_1),\ldots$ stabilizes. Thus, on a neighborhood of any specified $(x,v)$, for some $q$,

$$g_{q+1}=\sum_{j=0}^{q}a_j(x,v)g_j$$

with analytic coefficients. Along $x+tv$, the vector $(g_0,\ldots,g_q)$ satisfies a finite homogeneous linear differential system. If it is zero at $t=0$, uniform zero-solution uniqueness makes it zero for all sufficiently small $t$. Conversely, vanishing of $h(x+tv)$ on an interval makes all these initial jets zero. Hence the incidence of analytic line germs in the chosen submanifold is locally an analytic set defined by finitely many jet equations. This proves local finiteness of the equations; an infinite jet intersection has not been treated as automatically analytic.

Here is the exact compact-piece application of uniformization. Let $U$ be the open set on which $h^{-1}(0)$ is the chosen analytic submanifold, and let $V$ be one analytic chart of unit directions. In the analytic manifold $U\times V$, let $\mathcal I$ be the set of pairs $(x,v)$ for which $h(x+tv)=0$ for all sufficiently small real $t$. The finite-jet argument just given makes $\mathcal I$ locally analytic, hence locally closed and locally subanalytic. Choose countably many closed coordinate boxes $Q_\ell$, with interiors covering $U\times V$, each compactly contained in a neighborhood on which one finite list of jet equations defines $\mathcal I$. Such boxes come from a countable coordinate basis, by using rational smaller boxes inside these neighborhoods. Then

$$C_\ell=\mathcal I\cap Q_\ell$$

is compact, because it is the common zero set of finitely many continuous functions in the compact box. It is subanalytic in $U\times V$: the defining equations and box inequalities are semianalytic on a neighborhood of the box, and outside the box a neighborhood misses it. These pieces cover $\mathcal I$, including points on their boundaries.

The compact uniformization theorem therefore gives, for each nonempty $C_\ell$, a compact finite-dimensional real analytic manifold $W_\ell$ and an analytic map $u_\ell:W_\ell\to U\times V$ whose image is exactly $C_\ell$. This map is proper; it need not be injective or an immersion. Write its coordinate maps as $x:W_\ell\to U$ and $v:W_\ell\to S^{n-1}$, the latter obtained from the unit-direction chart. Neither the dimension nor the number of preimages of an incidence parameter needs to be controlled in the argument below. For a fixed $\ell$, put $W=W_\ell$ and define

$$F:W\times\mathbb R\longrightarrow\mathbb R^n, \qquad F(w,s)=x(w)+sv(w).$$

We need a neighborhood statement, not just a separate interval for each $w$. Fix $w_0\in W$. Near $(x(w_0),v(w_0))$, the finite-jet relation holds with analytic coefficients on an open parameter neighborhood. Shrink around this point to choose $\epsilon>0$ and a neighborhood $O\subset W$ of $w_0$ such that $(x(w)+sv(w),v(w))$ stays in that parameter neighborhood whenever $w\in O$ and $|s|<\epsilon$. This is possible by continuity, after choosing a smaller coordinate box whose closure lies in the relation’s domain; the directions have norm one. Every initial jet vector vanishes because $u_\ell(O)\subset\mathcal I$. The finite homogeneous differential system and zero-solution uniqueness then give $h(F(w,s))=0$ throughout $O\times(-\epsilon,\epsilon)$. Thus $F$ maps this open set into the chosen embedded analytic submanifold of dimension less than $n$, and its differential there has rank less than $n$.

At a fixed $w$, choose local coordinates on $W$. The differential columns of $F$ are $dx+s\,dv$ and $v$. Every $n$-rowed minor is a polynomial in $s$, vanishing on an interval about zero and therefore identically zero. If there are fewer than $n$ columns, the rank bound is automatic. Consequently $\operatorname{rank}dF<n$ for every real $s$, including after the line leaves $U$. No interval common to all source manifolds is asserted or needed.

Alternative using the proved partition theorem. Instead of compact uniformization, apply the compatible analytic partition theorem to the single locally subanalytic set $\mathcal I\subset U\times V$. The partition members contained in $\mathcal I$ form a countable collection of embedded analytic manifolds covering it. Their inclusions give the analytic maps $x,v$. On each such manifold the same parameter-neighborhood and polynomial-minor argument applies. This route gives the same rank bound and the same Sard arguments below; it requires neither compactness of these parameter manifolds nor properness of their inclusions. The compact uniformization route remains available with its stronger proper-source conclusion.

Theorem 2.1 and Corollary 2.2 of Regular values and degree modulo two supply the precise Sard theorem used here: the critical values of a smooth map between finite-dimensional Hausdorff second-countable manifolds are null in every target chart. Its proof uses the smooth inverse function theorem, Taylor estimates and Euclidean measure theory; it does not use a triangulation or the uniformization being constructed here. Our analytic map $F$ is smooth and every point of its domain is critical by the rank calculation. Thus its whole image in $\mathbb R^n$ is null.

The passage from nullness to meagreness uses compact pieces, not an assertion that every null set is meagre. A Hausdorff second-countable manifold admits a countable cover by compact coordinate balls: every point has a coordinate ball whose closed ball stays inside the coordinate domain, and second countability gives a countable subcover of their interiors. For completeness, to reduce any open cover to a countable one, choose for each basis element contained in some cover member one such member. These choices cover every point. The closures of the selected coordinate balls are compact because the closed Euclidean balls are compact and their inverse coordinate maps are continuous. Apply this to $W$, and take products with the intervals $[-j,j]$, $j\geq1$, to cover $W\times\mathbb R$ by countably many compact sets $K_i$. Each $F(K_i)$ is compact and therefore closed in $\mathbb R^n$. As a subset of the null image it contains no nonempty open set. It is consequently nowhere dense, and their countable union is precisely $F(W\times\mathbb R)$. This proves the assertion about all points on singular lines without assuming $W$ compact or $F$ proper.

For directions, suppose $w\mapsto[v(w)]$ had surjective differential somewhere. A local analytic section over an open direction chart would give $x=x(v)$. For the resulting map $(v,s)\mapsto x(v)+sv$, the determinant of its $n$ differential columns has leading coefficient

$$\det(\partial_1v,\ldots,\partial_{n-1}v,v)$$

in its polynomial in $s$. That coefficient is nonzero for a direction chart on the unit sphere. This contradicts the preceding rank calculation. The direction map therefore has only critical values. Apply the same Corollary 2.2 to the smooth map from $W$ to $\mathbb{RP}^{n-1}$. Its values are critical by the preceding argument. The critical image is null in every projective coordinate chart. Each compact image is closed because projective space is Hausdorff, and it has empty interior because an open coordinate ball is not null. The countable compact cover of $W$ therefore makes the image meagre in projective space. This applies in every dimension $n\geq2$; no compactness or properness of the whole direction map is required. In dimension one there are no line germs in a zero-dimensional set, so the assertion is immediate. Countable unions finish both claims. $\square$

In particular, inside any open simplex one can choose a centre outside a finite family of compact lower-dimensional sets and all their singular lines. One can also choose a parallel direction avoiding all singular directions of such a family. The precise category input is Theorem 3.2, the locally compact Hausdorff Baire theorem: no nonempty open subset of a locally compact Hausdorff space is contained in a countable union of nowhere dense sets. Apply it to the affine span of the simplex, identifying that span with Euclidean space, and to projective direction space. Coordinate balls with compact closure show that these spaces are locally compact; both are Hausdorff. Each compact lower-dimensional subanalytic set is closed with empty interior in the relevant affine span, hence nowhere dense, and the singular-line and direction sets have the countable nowhere-dense covers just proved. Their union cannot contain any prescribed nonempty open neighbourhood. This gives both choices. In affine dimension one the singular-line family is empty as above; affine dimension zero has no nonempty lower-dimensional set or direction choice.

Completing a finite-fibre set so its projection is open

For $p:\mathbb R^d\times\mathbb R\to\mathbb R^d$, call a set projection-open when the restricted map to $\mathbb R^d$ is open. All sets prescribed for compatibility in the following assertion are subanalytic subsets of the ambient Euclidean space; arbitrary subsets are not allowed.

Finite graph assertion. Suppose $A\subset\mathbb R^d\times\mathbb R$ is compact subanalytic, every fibre $A_x=\{y:(x,y)\in A\}$ is finite, and $A_1,\ldots,A_k\subset A$ are subanalytic. There is a finite analytic cell partition of $p(A)$ such that, over each base cell $C$,

$$A\cap p^{-1}(C)=\coprod_{i=1}^{m(C)}G_{C,i},\qquad G_{C,i}=\{(x,f_{C,i}(x)):x\in C\},\qquad f_{C,1}<\cdots<f_{C,m(C)}.$$

Every $f_{C,i}:C\to\mathbb R$ is analytic and globally subanalytic. Each graph is wholly inside or wholly outside every $A_j$. The projection $G_{C,i}\to C$ is an analytic diffeomorphism, with inverse $x\mapsto(x,f_{C,i}(x))$.

Proof of the graph assertion. The sets $A,A_1,\ldots,A_k$ are bounded. They are locally subanalytic at every point of their closures, because subanalyticity is assumed in the ambient space, including points not belonging to the sets. The bounded-chart comparison, part 2 therefore makes them globally subanalytic. Apply the finite analytic cell theorem simultaneously to these sets, in the coordinate order $x_1,\ldots,x_d,y$.

Each cell contained in $A$ is a graph or a band in the last coordinate over one cell of the induced decomposition of $\mathbb R^d$. A nonempty band contains an interval over every point of its base, contradicting finite fibres. Thus only graph cells occur. Over a fixed base cell $C\subset p(A)$, list the graph cells belonging to $A$ in their order in the cylindrical stack. Their analytic endpoint functions are defined on all of $C$, are strictly ordered there, and account for every point of every fibre over $C$. Compatibility of the decomposition with each $A_j$ gives the stated membership property. Finiteness of the decomposition gives finitely many base cells and graphs. If $A$ is empty there are none; if $d=0$, the graphs are its finitely many points. $\square$

The covering interpretation and its hypotheses. Fix $C$ and write $E=A\cap p^{-1}(C)$. The restriction $p:E\to C$ is proper: for a compact set $K\subset C$, its inclusion into $\mathbb R^d$ has compact image, and $A\cap p^{-1}(K)$ is a closed subset of the compact set $A$. This proves properness relative to $C$, even when $C$ is not closed in $\mathbb R^d$.

To verify the covering property, take $x_0\in C$. When there is more than one sheet, choose disjoint open intervals about the finitely many numbers $f_{C,i}(x_0)$. Continuity gives a relative open neighbourhood $V\subset C$ of $x_0$ on which every $f_{C,i}$ stays in its chosen interval. The sets $G_{C,i}\cap p^{-1}(V)$ are then disjoint relatively open subsets of $E$, cover $p^{-1}(V)\cap E$, and map analytically diffeomorphically to $V$. With one sheet this is immediate. Hence $p:E\to C$ is a finite analytic covering. The global real-height order identifies its sheets on all of $C$, so there is no permutation of sheets on going around a loop.

The rank argument remains valid: on a constant-rank analytic partition member, a rank smaller than its dimension would produce a positive-dimensional fibre by the analytic constant-rank theorem. What cannot be inferred from compactness alone is the covering property before the base has been suitably partitioned. For example, $\{(x,y):x=y^2,\ -1\le y\le1\}\to[0,1]$ is proper with finite fibres, but the fibre cardinality changes at zero. The cell proof supplies the required base partition as well as the ordered sheets.

Local compact enlargement. For every specified $a\in A$, there is a compact subanalytic set $B\supset A$, still with finite vertical fibres, and an ambient open neighbourhood $U$ of $a$ such that

$$p:B\cap U\longrightarrow\mathbb R^d$$

is open. This is a local assertion near $a$, not global openness of the projection of a nonempty compact set onto $\mathbb R^d$.

Proof by induction on $d$. For $d=0$, take $B=A$: every map to the one-point space is open. Now let $d\ge1$, and apply the graph assertion. Let $L$ be the union of the graph cells of dimension less than $d$, and set

$$A_2=\overline L.$$

The closure is taken in $\mathbb R^{d+1}$. Since $A$ is closed, $A_2\subset A$; it is compact. The Boolean and closure calculus and the dimension/frontier theorem, equations (D4) and (D12), show that $A_2$ is subanalytic and $\dim A_2<d$. This also covers $L=\varnothing$, when $A_2=\varnothing$. The set $A_2$ need not be disjoint from all the other graph cells. What is needed is that every point of $A\setminus A_2$ lies on a graph over a $d$-dimensional base cell. Such a base cell is open in $\mathbb R^d$, so its graph projects openly.

If $a\notin A_2$, choose $U$ disjoint from the closed set $A_2$ and take $B=A$. Given any relatively open $O\subset A\cap U$ and $b\in O$, intersect $O$ with a full-dimensional graph through $b$. Its projection is an open neighbourhood of $p(b)$ contained in $p(O)$. Taking the union over $b$ proves that $p(O)$ is open.

It remains to treat $a\in A_2$. The compact subanalytic set $p(A_2)$ has dimension less than $d$, by the image inequality (D5) in the same provider. Choose a parallel direction nonsingular for $p(A_2)$, using the preceding singular-direction argument, and use an orthogonal change of base coordinates to make this the $x_d$-direction. The vertical height coordinate $y$ is unchanged. Every line parallel to this direction meets $p(A_2)$ in finitely many points: its intersection is a compact subanalytic subset of a line, hence a finite union of points and intervals by the cell theorem; an interval would give a forbidden line germ in $p(A_2)$.

Write $x'=(x_1,\ldots,x_{d-1})$ and put

$$p_2:\mathbb R^{d-1}\times\mathbb R\times\mathbb R \longrightarrow\mathbb R^{d-1}\times\mathbb R, \qquad p_2(x',x_d,y)=(x',y).$$

The set $D=p_2(A_2)$ is compact and subanalytic. For fixed $x'$, only finitely many $x_d$ occur in $p(A_2)$, and each of these base points has only finitely many heights in $A_2\subset A$. Thus $D$ has finite vertical fibres over $\mathbb R^{d-1}$. Apply the induction hypothesis to $D$ at $p_2(a)$. It gives a compact subanalytic $B_2\supset D$, with finite fibres, and an open neighbourhood $V$ of $p_2(a)$ on which

$$r:B_2\cap V\longrightarrow\mathbb R^{d-1},\qquad r(x',y)=x',$$

is open.

Choose $R>0$ with $A_2\subset\operatorname{int}\overline{\mathbb B}_R$, and define

$$B'=p_2^{-1}(B_2)\cap\overline{\mathbb B}_R, \qquad B=A\cup B'.$$

These sets are subanalytic. The set $B'$ is closed and bounded, hence compact, and contains $A_2$. At a fixed base point $(x',x_d)$, its possible heights belong to the finite fibre $(B_2)_{x'}$. Consequently $B$ is compact and has finite vertical fibres as well.

Take an ambient open neighbourhood $U$ of $a$ contained in $\operatorname{int}\overline{\mathbb B}_R\cap p_2^{-1}(V)$. On this neighbourhood the ball has no effect. After reordering the coordinates, $p_2^{-1}(B_2\cap V)$ is $(B_2\cap V)\times\mathbb R$, and its base projection is $r\times\operatorname{id}_{\mathbb R}$. This product map is open: an open set is a union of open product sets, whose images are products of an open image under $r$ with an open interval. Its restriction to the relatively open subset cut out by $U$ remains open. Therefore $p:B'\cap U\to\mathbb R^d$ is open.

Finally let $O$ be relatively open in $B\cap U$, and take $b\in O$. If $b\in B'$, the relatively open set $O\cap B'$ supplies an open base neighbourhood of $p(b)$ inside $p(O)$. If $b\notin B'$, then $b\in A\setminus A_2$, because $A_2\subset B'$. A full-dimensional graph through $b$ supplies that neighbourhood exactly as in the earlier case. Thus $p(O)$ is open. This completes the induction. $\square$

Compact radial completion by doubling

Let $a\in\mathbb R^n$, let $A\subset\mathbb R^n\setminus\{a\}$ be compact subanalytic, and suppose the radial map

$$\rho_a:\mathbb R^n\setminus\{a\}\longrightarrow S^{n-1}, \qquad \rho_a(x)=\frac{x-a}{|x-a|},$$

has finite fibres on $A$. For every $b\in A$, there are a compact subanalytic set $D\subset\mathbb R^n\setminus\{a\}$ and an ambient open neighbourhood $W$ of $b$ with these properties: $A\cap W\subset D$; the fibres of $\rho_a|_D$ are finite; and $\rho_a:D\to S^{n-1}$ is open. Moreover, a finite partition of $D$ into analytic radial graphs can record membership in $A$ and any prescribed finite subanalytic family. Thus, near $b$, the original set is a union of the selected graph pieces restricted to $W$. The completion need not contain all of $A$.

If $n=1$, take $D=A$. There are only two directions, with the discrete topology, so every restricted radial map is open, and finite radial fibres make $A$ finite. We now treat $n\ge2$. Translation, rotation and positive scaling let us take $a=0$ and $b=e_n$, without changing any of the required properties. Write

$$v(z)=(z,\sqrt{1-|z|^2}),\qquad |z|<1, \qquad (z,r)\longmapsto r\,v(z),\quad r>0.$$

These are analytic polar coordinates near $e_n$; the inverse sends $x$ to $((x/|x|)',|x|)$. The radial map is projection onto $z$, followed by the analytic sphere chart $v$.

A compact affine input. Choose a small closed coordinate box $Q\subset\{|z|<1\}$ with $0$ in its interior. Restrict $A$ to directions in $v(Q)$ and take its polar-coordinate image $A_Q$. Since $A$ is compact and avoids zero, its radii lie in some compact interval $[r_-,r_+]\subset(0,\infty)$. Therefore $A_Q\subset Q\times[r_-,r_+]$ is compact subanalytic, with finite fibres over $z$. The polar coordinate maps are analytic on neighbourhoods of the compact sets in question; the bounded-chart and compact-image calculus used above justifies these subanalytic images. In particular, this step makes no assertion about an unbounded analytic image.

Apply the preceding local compact enlargement to $A_Q$ at $(0,1)$. Obtain a compact subanalytic set $E\supset A_Q$ with finite vertical fibres, and an open neighbourhood $N$ of $(0,1)$, on which $\pi:E\cap N\to\mathbb R^{n-1}$, $\pi(z,r)=z$, is open. We use only the part of $E$ inside the polar coordinate domain and at positive radii.

Choosing the band before the cap. Choose a small product neighbourhood of $(0,1)$ whose closure lies in $N\cap(\operatorname{int}Q\times(0,\infty))$. Inside its radial interval choose $\alpha<1<\beta$ so that neither endpoint belongs to the finite fibre $E_0$. The two sets

$$F_\alpha=\pi\bigl(E\cap(\mathbb R^{n-1}\times\{\alpha\})\bigr),\qquad F_\beta=\pi\bigl(E\cap(\mathbb R^{n-1}\times\{\beta\})\bigr)$$

are compact, and neither contains zero. Consequently there is an open base neighbourhood $V$ of zero, with compact closure in $\operatorname{int}Q$, such that

$$\overline V\times[\alpha,\beta]\subset N, \qquad \overline V\cap(F_\alpha\cup F_\beta)=\varnothing.$$

Thus no fibre of $E$ over $\overline V$ meets either radial endpoint. Put $E^\circ=E\cap(V\times(\alpha,\beta))$. It is relatively open in $E\cap N$, so $\pi:E^\circ\to\mathbb R^{n-1}$ is open. Its image contains zero. Choose $c\in(0,1)$, sufficiently close to one, so that the closed spherical cap

$$K=\{v\in S^{n-1}:v_n\ge c\}$$

has its coordinate image contained in $V\cap\pi(E^\circ)$. Define

$$E_K=\{r v(z):(z,r)\in E,\ v(z)\in K,\ \alpha\le r\le\beta\}.$$

This is a compact subanalytic set with finite radial fibres and radii in $[\alpha,\beta]$, none equal to either endpoint. It projects onto $K$. Its projection is open relative to $K$: in polar coordinates it is the restriction of $E^\circ\to\mathbb R^{n-1}$ over the closed base set corresponding to $K$. If $O$ is relatively open in this restriction, write it as the intersection of the restriction with a relatively open $O'\subset E^\circ$. Then its image is $\pi(O')\cap v^{-1}(K)$, which is relatively open in $v^{-1}(K)$. This is the elementary base-change property for an open map; the radial endpoint exclusion is what makes the closed-band restriction equal to this restriction of $E^\circ$.

Reflection. Put $k=(1-c)/(1+c)>0$. For $v=(v',v_n)\in S^{n-1}$, define

$$J(v',v_n)=\left( \frac{2k\,v'}{1-v_n+k^2(1+v_n)}, \frac{1-v_n-k^2(1+v_n)}{1-v_n+k^2(1+v_n)} \right).$$

The denominator is at least $2\min(1,k^2)>0$. With $u=1-v_n$ and $w=1+v_n$, the sphere equation gives $|v'|^2=uw$, and

$$4k^2|v'|^2+(u-k^2w)^2=(u+k^2w)^2.$$

Hence $J$ is an analytic map of the sphere to itself. Away from the south pole, use the stereographic coordinate $s=v'/(1+v_n)$. For $s\ne0$ its transformed coordinate is

$$s\longmapsto\frac{k s}{|s|^2}.$$

This inversion is an involution, fixes $|s|^2=k$, and interchanges $|s|^2<k$ with $|s|^2>k$. The displayed sphere formula interchanges the two poles and is analytic there as well. Thus $J^2=\operatorname{id}$ everywhere. Since $|s|^2=(1-v_n)/(1+v_n)$, the fixed sphere is exactly $v_n=c$; $J$ fixes that sphere pointwise and exchanges the interiors of $K$ and $K^*=\{v_n\le c\}$. This proves the stated reflection properties, including the inversion radius $\sqrt{k}$.

Extend it radially by $\widehat J(rv)=rJ(v)$ for $r>0$, and put

$$D=E_K\cup\widehat J(E_K).$$

The radial extension is analytic away from zero. Both sets in the union are compact subanalytic and avoid zero. A fibre of $D$ over a direction $v$ has radii in the union of the two finite radius sets of $E_K$ over $v$ and over $J(v)$. Thus every radial fibre remains finite.

Openness on the doubled set. The map on $E_K$ is open relative to $K$; conjugating by $\widehat J$ and $J$ gives the corresponding assertion for $\widehat J(E_K)$ over $K^*$. For a relatively open $O\subset D$ and a point $x\in O$ whose direction is in the interior of one cap, the applicable relative openness supplies an open sphere neighbourhood of $\rho_0(x)$ in $\rho_0(O)$.

At a boundary direction $v_n=c$, the reflection fixes every point of the ray, so a point $x\in D$ above that direction belongs to both copies. If $x\in O$, their relative openness gives a relative neighbourhood $U_+\subset K$ and a relative neighbourhood $U_-\subset K^*$ of $v=\rho_0(x)$, both contained in $\rho_0(O)$. Write $U_+=V_+\cap K$ and $U_-=V_-\cap K^*$ with $V_+,V_-$ open in the sphere. Then $V_+\cap V_-$ is a sphere neighbourhood of $v$ contained in $U_+\cup U_-$, because $K\cup K^*=S^{n-1}$. This also lies in $\rho_0(O)$. Every point of that image consequently has an open neighbourhood within it, proving that $\rho_0:D\to S^{n-1}$ is open globally.

Local membership and graph pieces. In the open neighbourhood

$$W=\{r v:v\in\operatorname{int}K,\ \alpha<r<\beta\}$$

of $e_n$, every point of $A$ belongs to $E_K\subset D$: its direction is in $v(Q)$, its polar coordinates belong to $A_Q\subset E$, and it satisfies the defining cap and radius conditions. The reflected copy lies over $K^*$, so it adds no points over $\operatorname{int}K$. Additional points supplied by the original affine enlargement may remain, as intended.

To obtain the promised finite graph partition, cover the sphere by finitely many closed semialgebraic angular patches contained in analytic coordinate charts; for example, use $\{\pm v_i\ge1/\sqrt n\}$ and the charts on $\{\pm v_i>0\}$. The polar images of $D$ over these closed patches are compact subanalytic sets with finite height fibres. Apply the preceding finite graph assertion on each patch, requiring compatibility with the images of $D\cap A$, the prescribed subanalytic subsets, and both cap boundaries. These bounded images are globally subanalytic by the bounded-chart comparison. Transfer the resulting finitely many base pieces to the sphere and take a common finite analytic cell refinement, compatible also with the angular patches. Assign each refined base cell to one patch containing it. Restrict that patch’s graph functions to the refined cell. Their domains are the whole refined cell, the functions are analytic and strictly ordered, and they account for all points of $D$ over it. Each resulting graph is wholly inside or wholly outside $A$ and every prescribed subset. In particular, $A\cap W$ is the union of the selected graphs restricted to $W$. Undoing the initial similarity proves the assertion for the original $a$ and $b$. $\square$

Extending the skeleton without changing old simplices

We prove the relative theorem by induction on $m=\dim K$. The assertion for $m=0$ is immediate. Suppose it is proved in lower dimensions. For an old $m$-simplex $\sigma$, work in its $m$-dimensional affine span; boundaries and interior in that span will be distinguished from the old skeleton.

Reduction to closed obstacles. For a subanalytic subset $A\subset\sigma$, put $B_0=A$ and $B_{j+1}=\overline{B_j}\setminus B_j$, taking closures in the affine span. Each closure stays in the compact set $\sigma$. The strict frontier theorem (D12) gives $\dim B_{j+1}<\dim B_j$ whenever $B_j\ne\varnothing$. Thus the sequence reaches the empty set after finitely many steps. Write $C_j=\overline{B_j}$. The identities $B_j=C_j\setminus B_{j+1}$, substituted backwards from the last nonempty term, express membership in $A$ by a finite Boolean combination of the compact closed subanalytic sets $C_j$.

If $\dim C_j<m$, retain $C_j$ itself as an obstacle. If $\dim C_j=m$, retain its boundary in the affine span. This boundary has dimension less than $m$ by (D15). Its complement is the disjoint union of the two open sets $\operatorname{int}C_j$ and $\operatorname{aff}(\sigma)\setminus C_j$. Hence a connected set avoiding the boundary lies wholly in one or the other. Once the lower-dimensional obstacles and the membership data on the old skeleton are respected, this observation recovers membership in all the $C_j$, and hence in $A$. Each compact old simplex meets only finitely many members of the original locally finite family, so only finitely many obstacles are needed on $\sigma$.

A finite completed set. Denote these compact lower-dimensional obstacles by $F_1,\ldots,F_q$. Choose $a\in\sigma^\circ$ outside their union and all their singular lines, by the preceding singular-line and Baire arguments. Every ray from $a$ meets each $F_j$ in finitely many points. Let $p_a$ send a point of the affine span other than $a$ to the unique point where its positive ray from $a$ meets $\partial\sigma$. Radial direction identifies the sphere in that affine span homeomorphically with $\partial\sigma$: every ray has exactly one boundary point, and the inverse direction map is a continuous bijection from the compact boundary to the Hausdorff sphere. Thus openness of the direction map is equivalent to openness of $p_a$.

At each point of each $F_j$, apply the compact radial completion. Choose its graph partition to record membership in every $F_l$. Compactness of the finitely many obstacles gives finitely many completion neighbourhoods covering their union. Let $D_1,\ldots,D_N$ be the corresponding completed compact sets and put $C=\bigcup_iD_i$. Then $C$ is compact, avoids $a$, has finite radial fibres, and contains every $F_j$. The map $p_a:C\to\partial\sigma$ is open: for a relatively open $O\subset C$, each $O\cap D_i$ is relatively open in $D_i$, so $p_a(O)=\bigcup_i p_a(O\cap D_i)$ is open. If there are no obstacles, use $C=\varnothing$.

Choose $0<\delta<1$ only after $C$ has been formed. If $C\ne\varnothing$, set $R=\max_{x\in\sigma}|x-a|>0$ and require

$$\delta R<\operatorname{dist}(a,C).$$

If $C=\varnothing$, any $\delta\in(0,1)$ works. Define the compact subanalytic set

$$\Gamma=(C\cap\sigma)\ \cup\ \partial\sigma\ \cup\ \bigl(a+\delta(\partial\sigma-a)\bigr).$$

For a point $x=a+t(u-a)$ with $u\in\partial\sigma$ and $x\in C\cap\sigma$, the inequality $|x-a|>\delta R\ge\delta|u-a|$ gives $t>\delta$. Consequently every fibre of $\Gamma$ has least height $\delta$, greatest height $1$, and finitely many heights between them. The open inner simplex contains no point of $\Gamma$. Choosing $\delta$ by the distance from the original obstacles alone would not give this conclusion for their added completions.

Why clipping retains openness. Take a relatively open $O\subset\Gamma$ and $x\in O$. If $x$ is on the outer boundary graph $\partial\sigma$, its intersection with $O$ projects openly by the identity map of $\partial\sigma$. If $x$ is on the inner boundary graph, use the homeomorphism $a+\delta(u-a)\mapsto u$. In either case this supplies a neighbourhood of $p_a(x)$ in $p_a(O)$.

Every other point of $\Gamma$ lies in $C\cap\sigma^\circ$. Choose an ambient open neighbourhood $U$ of $x$ contained in $\sigma^\circ$ and small enough that $C\cap U\subset O$; this is possible from the relative openness of $O$ in $\Gamma$. Since $C\cap U$ is relatively open in $C$, its image is an open neighbourhood of $p_a(x)$ contained in $p_a(O)$. These cases prove that $p_a:\Gamma\to\partial\sigma$ is open. In particular, the argument does not assume that cutting an open map by an arbitrary closed subset of its domain preserves openness.

Recorded membership. Take a finite radial graph partition of $\Gamma$, refined to respect the intersections with the finitely many graph pieces of all the $D_i$, and the two boundary graphs. Suppose a refined piece $P$ meets $F_j$ at $x$. A completion neighbourhood covering $x$ supplies a graph piece $G\subset D_i$ containing $x$ and lying in $F_j$, because that partition recorded membership in $F_j$. Compatibility with $G\cap\Gamma$ implies $P\subset G\cap\Gamma\subset F_j$. Thus each refined piece is wholly inside or outside every obstacle.

The equivalent local-to-global criterion is also useful: on a connected piece, its intersection with a closed obstacle is closed; if the recorded completion neighbourhoods make membership locally constant at all points of that intersection, it is open as well. Connectedness then makes it empty or the whole piece. Here the compatible finite graph refinement supplies precisely that local constancy, rather than assuming it from closedness alone.

Project the graph pieces and their relevant frontiers to the old skeleton, and include the original membership data there. These images are subanalytic by the bounded Boolean and projection calculus. Indeed $\Gamma$ is compact and avoids $a$, and the normalized radial projection to a linear simplex has a semialgebraic graph; its restriction to these bounded pieces has a globally subanalytic graph. This also treats pieces which are not closed; properness of a restriction to a nonclosed piece is not being assumed. Only finitely many data sets are contributed by each old simplex, and all of them lie on its boundary. Local finiteness of the old complex therefore makes the resulting family locally finite. The induction hypothesis yields a subdivision $H$ of the skeleton and a subanalytic homeomorphism $\tau$ which preserves each old face setwise, is analytic on every new open simplex, and is compatible with all this projected data.

Ordered branches, including their values on faces

Fix $\sigma$ and its completed set $\Gamma$. Pull the radial graph back by the boundary homeomorphism, retaining normalized height as a separate coordinate:

$$\Lambda=\{(u,t)\in\partial\sigma\times[\delta,1]: a+t(\tau(u)-a)\in\Gamma\}.$$

It is compact. The map $(u,t)\mapsto a+t(\tau(u)-a)$ from $\Lambda$ to $\Gamma$ is a continuous bijection: the radial base point and the positive normalized height uniquely determine $u$ and $t$. As its domain is compact and its target Hausdorff, it is a homeomorphism. Under it, the projection $\Lambda\to\partial\sigma$ corresponds to $\tau^{-1}\circ p_a$. It is consequently open and has finite fibres.

Over an open simplex $\eta^\circ$ of $H$ in $\partial\sigma$, the graph partition gives all the finitely many strictly ordered heights

$$\delta=r_0(u)<r_1(u)<\cdots<r_s(u)=1, \qquad u\in\eta^\circ.$$

Each function is analytic there. The recorded projected data ensure that each graph occurs over the whole open simplex or not at all; no branch has merely a proper subset as its domain. Analyticity follows by restricting a graph chart and composing with $\tau|_{\eta^\circ}$. The normalized radial coordinate is analytic on each old face cone, and $\tau$ preserves that old face. When $\eta$ is a vertex, the functions are simply its fibre heights.

A single limit at every face point. For a positive-dimensional $\eta$, fix one branch $r_i$ and $u_0\in\overline\eta$. For integers $l\ge1$, let

$$H_l=\overline{\{(u,r_i(u)): u\in\eta^\circ,\ |u-u_0|<1/l\}},$$

taking the closure in the compact space $\Lambda\cap(\overline\eta\times[\delta,1])$. The domain inside the braces is nonempty and convex, hence connected, and its graph is connected because $r_i$ is continuous. Its closure $H_l$ is therefore connected. These are nested nonempty compact sets, so their intersection $H_\infty$ is nonempty. All of its points have base coordinate $u_0$, and that fibre of $\Lambda$ is finite.

This intersection has only one point. Otherwise choose pairwise disjoint open neighbourhoods of its finitely many points. Some $H_l$ must be contained in their union: if none were, the nested nonempty compact sets obtained by intersecting $H_l$ with the complement of that union would have a common point outside the union, contrary to its covering $H_\infty$. But every $H_l$ contains $H_\infty$, so that $H_l$ would meet at least two of the disjoint neighbourhoods, contradicting its connectedness. Write the unique point as $(u_0,\overline r_i(u_0))$.

Thus every sequence $u_j\in\eta^\circ$ converging to $u_0$ has $r_i(u_j)\to\overline r_i(u_0)$. If not, a subsequence of the bounded heights staying a positive distance from that value would have a convergent subsequence, producing another point of $H_\infty$. This also shows that the closure of the entire graph of $r_i$ has exactly one point over every $u_0\in\overline\eta$. Its projection onto $\overline\eta$ is a continuous bijection from a compact space to a Hausdorff space, hence a homeomorphism: closed subsets of the compact domain have compact, therefore closed, images. Its inverse proves that $\overline r_i$ is continuous on the whole closed simplex, including sequences travelling through different faces. Its graph is subanalytic by the closure calculus. The endpoint branches extend as the constants $\delta$ and $1$.

Which branch occurs on a face. Let $\theta$ be a face of $\eta$. Over $\theta^\circ$, list the graph heights as $g_0<\cdots<g_t$. At each $u\in\theta^\circ$, the value $\overline r_i(u)$ is one of these heights because its graph lies in $\Lambda$. Its index is locally constant. At any fixed point the finite distinct heights have positive gaps, and continuity of $\overline r_i$ and all the $g_j$ preserves a separating interval for the chosen height on a sufficiently small neighbourhood. Only that same $g_j$ can equal $\overline r_i$ there. The connectedness of $\theta^\circ$ makes the index constant over the entire open face. This argument also covers a zero-dimensional face.

No face branch can be skipped. Two consecutive heights $r_i,r_{i+1}$ retain their weak order on $\overline\eta$. Suppose their restrictions on $\theta^\circ$ are distinct, and at some $u_0\in\theta^\circ$ another height $g_j(u_0)$ lies strictly between their limiting values. Choose $\epsilon>0$ so that the interval $(g_j(u_0)-\epsilon,g_j(u_0)+\epsilon)$ is strictly between those two values, even after each endpoint is moved by $\epsilon$. The relatively open subset