Approximating Continuous Function by Smooth Functions on Orbit Spaces

: In this paper, we study the approximation of continuous functions on a subclass of singular space—the subcartesian space. As is well known, the orbit space of the proper action of a Lie group on a smooth manifold is a subcartesian space. We prove that continuous functions on the orbit space can be approximated by smooth functions.


Introduction
There has long been perceived the need for an extension of the framework of smooth manifolds in differential geometry, which is too restrictive and does not admit certain basic geometric intuitions.Sikorski's [1] theory of differential spaces studies the differential geometry of a large class of singular spaces, which both contains the theory of manifolds and allows the investigation of singularities.Analogous to algebraic geometry, which is the investigation of geometry in terms of polynomials, the theory of differential space is the investigation of geometry in terms of differentiable functions.
Precisely, a differential structure on a topological space S is a family C ∞ (S) of realvalued functions on S satisfying the following conditions: 1.
The family { f −1 (I)| f ∈ C ∞ (S)and I is an open interval in R} is a subbasis for the topology of S.

2.
If If f : S → R is a function such that, for every x ∈ S, there exists an open neighborhood U of x, and a function f x ∈ C ∞ (S) satisfying Here, the subscript vertical bar | denotes a restriction.(S, C ∞ (S)) is said to be a differential space.Functions f ∈ C ∞ (S) are called smooth functions on S.
It follows that a smooth manifold M can be characterized as a differential space (M, C ∞ (M)), with C ∞ (M) being all smooth functions on the smooth manifold M, such that every point has a neighborhood U diffeomorphic to an open subset V of R n , where n is the dimension of the manifold, the differential structures on U and V are generated by restrictions of smooth functions of M and R n , respectively, and diffeomorphism is in the sense of differential space.This definition can be weakened by not requiring V to be open in R n and allowing n to be an arbitrary non-negative integer.Definition 1.A differential space S is said to be subcartesian [2] if every point of S has a neighborhood U diffeomorphic to a subset of some Cartesian space R n .(U, Φ, R n ) is said to be a local chart of p, where Φ : U → Φ(U) ⊆ R n is the diffeomorphism.
The theory of subcartesian spaces has been developed by Śniatycki et al. in recent years.See [2] for a systematic treatment of this topic.From the above definition, any subset of a Euclidean space endowed with the differential subspace structure is a subcartesian space.Another typical example of subcartesian space is the orbit space of the proper action of a connected Lie group on a smooth manifold [2][3][4].
In this paper, we investigate the following problem: given any continuous function f on a subcartesian space S, we ask whether it can be approximated by smooth functions on S.
It is well known that a continuous function on a smooth manifold can be approximated by a smooth function [5,6], as stated by the following theorem.Theorem 1 ([5]).Let M be a smooth manifold and f : M → R be a continuous function.Then, for any δ > 0, there exists a smooth function h However, for a subcartesian space S, we cannot infer that a continuous function on S can be approximated by smooth functions on S. The first obstruction is that, given p ∈ S, with (U, Φ, R n ) being its local chart, we cannot infer that the continuous function f restricting to U can be extended to a continuous function f on an open subset of R n that contains In this paper, we investigate a special class of subcartesian space-orbit space R of the proper action of a connected Lie group G on a smooth manifold M. We overcome the obstructions described above by taking advantage of the geometric structure of the orbit space, which is obtained by the reduction of the symmetry of smooth manifolds.Precisely, we first investigate the local approximation problem, which is defined on an open neighborhood of p ∈ R, and then study passages from local to global.Since continuous or smooth functions on R can be lifted to G-invariant continuous or smooth functions on M, respectively, the local problem can be solved by approximating G-invariant continuous functions by G-invariant smooth functions on M.This can be solved by using the geometry of the symmetry of smooth manifolds together with Theorem 1.For the problem of passages from local to global, the geometric structure of the symmetry of smooth manifolds also plays a central role.We obtain the following theorem that gives a positive answer to the problem proposed above.Theorem 2. Let f : R → R be a continuous function on the orbit space R.Then, for any δ > 0, for any y ∈ R.
To the best of our knowledge, this is the first result on the approximation of functions in subcartesian space.We have not seen any approximation theorem in subcartesian space in the existing literature.
The paper is organized as follows.In Section 2, we recall some basic definitions in the subcartesian space.In Section 3, we recall some basic facts about the orbit space.In Section 4, we prove our main results.

Subcartesian Space Definition 2 ([2]
).Let S 1 and S 2 be two differential spaces.A map ϕ : An alternative means of constructing a differential structure on a set S is as follows.Let F be a family of real-valued functions on S. Endow S with the topology generated by a subbasis Clearly, F ∈ C ∞ (S).It is proven in [2] that C ∞ (S) defined here is a differential structure on S. We refer to it as the differential structure on S generated by F .Let S be a differential space with a differential structure C ∞ (S), and let T be an arbitrary subset of S endowed with the subspace topology (open sets in T are of the form T ∩ U, where U is an open subset of S).Let ).The space S(T) of restrictions to T ⊆ S of smooth functions on S generates a differential structure C ∞ (T) on T such that the differential-space topology of S coincides with its subspace topology.In this differential structure, the inclusion map i : T → S is smooth.
In other words, S(T) is the space of restrictions to T of smooth functions on S. Now, consider an equivalence relation ∼ on a differential space S with differential structure C ∞ (S).Let R = S/∼ be the set of equivalence classes of ∼, and let ρ : S → R be the map assigning to each x ∈ S its equivalence class ρ(s).

Definition 4 ([2]
).The space of functions on R, given by is a differential structure on R. In this differential structure, the projection map ρ : S → R is smooth.
It should be emphasized that, in general, the quotient topology of R = S/∼ is finer than the differential-space topology defined by C ∞ (R).
A condition for the differential-space topology to coincide with the quotient topology is given below.

Proposition 1 ([2]
).The topology of R induced by C ∞ (R) coincides with the quotient topology of R if, for each set U in R that is open in the quotient topology, and each y ∈ U, there exists a function f ∈ C ∞ (R) such that f (y) = 1 and f | R\U = 0, where R\U denotes the complement of U in R.

Orbit Space
Consider the smooth and proper action of a locally compact connected Lie group G on a manifold M. Recall that the action is proper if, for every convergent sequence (x n ) in M and a sequence (g n ) in G such that the sequence (g n x n ) is convergent, the sequence (g n ) has a convergent subsequence (g n k ) and lim k→∞ The isotropy group G x of a point x ∈ M is We endow the orbit space R = M/G with the quotient topology.In other words, a subset
In the following, we introduce the definition of a slice, which plays a central role in the geometric structure of the symmetry of smooth manifolds.

Definition 5 ([2]
).A slice through x ∈ M for an action of G on M is a submanifold S x of M containing x such that 1.
S x is transverse and complementary to the orbit Gx of G through x.In other words,

2.
For every x ′ ∈ S x , the manifold S x is transverse to the orbit Gx ′ ; in other words, 3. S x is G x -invariant.Specifically, gy ∈ S x for any g ∈ G x and y ∈ S x .

Let x
Given a G-invariant Riemannian metric k on M, we denote by verTM the generalized distribution on M consisting of vectors tangent to G-orbits in M, and by horTM the korthogonal complement of verTM.The existence of a slice through x ∈ M is ensured by the following result.

Proposition 3 ([2]).
There is an open ball B in horT x M centered at 0 such that S x = exp x B is a slice through x for the action of G on M, where exp x v is the value at 1 of the geodesics of the G-invariant Riemannian metric originating from x in the direction v. Further, the set GS x = {gq|g ∈ G and q ∈ S x } is a G-invariant open neighborhood of x in M.
Let H = G x .By construction, S x = exp x B, where exp x is an H-equivariant map from a neighborhood of 0 in T x M to a neighborhood of x in M, and B is a ball in horT x M invariant under a linear action of H centered at the origin.The action of H on T x M is linear, and it leaves horT x M invariant.Hence, it gives rise to a linear action of H on horT x M.Moreover, the restriction of exp x to B gives a diffeomorphism ψ : B → S x , which intertwines the linear action of H on horT x M and the action of H on S x .
Since B is an H-invariant open subset of horT x M and the action of H on horT x S x is linear, via the theorem of G. W. Schwarz [7], smooth H-invariant functions on S x are smooth functions of algebraic invariants of the action of H on horT x M. Let R[horT x M] H denote the algebra of H-invariant polynomials on horT x M. Hilbert's Theorem [8] ensures that R[horT x M] H is finitely generated.Let σ 1 , • • • , σ n be a Hilbert basis for R[horT x M] H consisting of homogeneous polynomials.The corresponding Hilbert map induces a monomorphism σ : where Hv is the orbit of H through v ∈ horT x M treated as a point in (horT x M)H.Let Q be the range of σ.By the Tarski-Seidenberg Theorem [9], Q is a semi-algebraic set in R n .Let be the bijection induced by σ. ϕ is a diffeomorphism [2].Since B is an H-invariant open neighborhood of 0 in horT x S x , it follows that B/H is open in (horT x M)/H.Hence, B/H is in the domain of the diffeomorphism ϕ : (horT x M)/H → Q, which induces a diffeomorphism of B/H onto ϕ(B/H) ⊆ Q ⊆ R n .Thus, B/H is diffeomorphic to a subset of R n .However, B/H is diffeomorphic to S x /H, and S x /H is diffeomorphic to GS x /G.Therefore, GS x /G is diffeomorphic to a subset of R n .Hence, we have the following.

Theorem 3 ([2]
).The orbit space R = M/G of a proper action of G on M with the differential structure C ∞ (R) is subcartesian.

Approximating Continuous Function on Orbit Space by Smooth Functions
In this section, we prove Theorem 2. We first study the local approximation problem.

Lemma 1 ([5]
).Let M be a smooth manifold and f be a continuous function on M. Given ϵ > 0, then there exists h ∈ C ∞ (M) such that |h(p) − f (p)| < ϵ, for p ∈ M.

Lemma 2 ([10]
).Let U, V be two open subsets of the smooth manifold M satisfying that cl(U) is compact and cl(U) ∩ cl(V) = ∅, where cl(U) and cl(V) denote the closure of U and V.Then, there exists a smooth function f Now, consider the subcartesian space (R = M/G, C ∞ (R)).The following result provides a positive solution to the local approximation of a continuous function on R by a smooth function.

Lemma 3.
For each y 0 ∈ R, there exists a local neighborhood V of y 0 satisfying that, for any continuous function f on V and any δ > 0, there exists a smooth function f for any y ∈ V.
Proof.Let x 0 ∈ M such that ρ(x 0 ) = y 0 .Let H be the isotropy group of x 0 and S x 0 = exp x 0 B be a slice through x 0 ∈ M, where exp x 0 is an H-equivariant map from a neighborhood of 0 in T x 0 M to a neighborhood of x 0 in M, and B is a ball in horT x 0 M invariant under the linear action of H centered at the origin.
Then, for any continuous function f on ρ(GS x 0 ), it follows that exp * x 0 (ρ * f | S x 0 ) is a continuous function on B, where ρ : M → R is the orbit map.From Lemma 1, we know that for any δ > 0, there exists a smooth function h ∈ C ∞ (B), such that where dµ(g) is the Haar measure on H normalized so that volH = 1.The set GS x 0 is a G-invariant open neighborhood of x 0 in M. We can define a Ginvariant smooth function f1 on GS x 0 as follows.For each x ′′ ∈ GS x 0 , there exists g ∈ G such that x ′′ = gx ′ for x ′ ∈ S x , and we set f1 (x ′′ ) = h(x ′ ).
f1 is well defined.Let x ′′ = g 1 x 1 , where g 1 ∈ G and x 1 ∈ S x .From the above definition, we have f1 (x ′′ ) = h(x 1 ).On the other hand, since g 1 x 1 = gx ′ , we have g −1 1 gx ′ = x 1 .Since x ′ , x 1 ∈ S x , it follows from Definition 5 and Proposition 3 that g −1 1 g ∈ H. Hence, h(x 1 ) = h(g −1 1 gx ′ ) = h(x ′ ) since h is H-invariant; this yields that f1 is well defined.From the definition of f1 , we know that f1 is G-invariant, which descends to a function f 1 on ρ(S x 0 ) such that ρ * f 1 = f1 .Moreover, for each y ∈ ρ(exp x 0 B), we have where It follows from Lemma 2 that there exists a smooth function η : horT x M → R such that which yields a smooth function η • exp −1 x on S x .Since V, W are H-invariant, then by averaging η • exp −1 x over S x , we obtain a G x -invariant smooth function η on S x satisfying that which can be extended to a smooth G-invariant function η1 on M. Now, consider the function η1 f1 on M. Since exp x V ⊆ GS x 0 , it follows that η1 f1 is a smooth G-invariant function on M satisfying that η1 f1 G exp x (cl(W)) = f1 G exp x (cl(W)) .Since η1 f1 descends to ) is a differential subspace of R.This completes the proof of the claim.Hence, for y 0 ∈ R and for x ∈ ρ −1 (y 0 ), there exists a local neighborhood ρ(GS x ) of y 0 satisfying that, for any continuous function f on ρ(GS x ) and any δ > 0, there exists a smooth function for any y ∈ ρ(GS x ).Hence, the result follows.
In the following, we investigate passages from local to global for the approximation problem on R.
open subsets of horT x M such that cl(W) ⊆ V and cl(V) are compact, where B satisfies that exp x B = S x , and cl(W) and cl(V) denote the closure of W and V. Let T be an open subset of R and (ρ(exp x U), ψ) be the local coordinate for R induced by the Hilbert map (2).Let f : R → R be a continuous map satisfying that f | T ∈ C ∞ (T), where (T, C ∞ (T)) is a differential subspace of (R, C ∞ (R)).Then, for any δ > 0, there exists a continuous map h : R → R, such that (1) h(y) = f (y), for any y ∈ R\ρ(exp x (V)); (2) h| T∪ρ(exp x W) ∈ C ∞ (T ∪ ρ(exp x W)); (3) |h(y) − f (y)| < δ, for all y ∈ R.
Proof.It follows from Lemma 2 that there exists smooth function η : horT which yields a smooth function η • exp −1 x on S x .Since V, W are H-invariant, then by averaging η • exp −1 x over S x , we obtain a G x -invariant smooth function η on S x satisfying that which can be extended to a smooth G-invariant function on M. Hence, we obtain a function η ∈ C ∞ (R) satisfying that η(y) = 1, y ∈ ρ(exp x (cl(W))), It follows from Lemma 3 that the function f | ρ(GS x ) can be approximated by smooth functions on ρ(GS x ).In other words, for any δ > 0, there exists a smooth function Since η(y) = 0, y ∈ R\ρ(exp x (V)), it follows that h(y) = f (y), for any y ∈ R\ρ(exp x (V)); Then, the result follows.
Lemma 5 ([5]).Let X be a second, countable, locally compact Hausdorff topological space.Then, there exist countable many sets G where cl(G j ) denotes the closure of G j , j = 1, 2, • • • .Lemma 6.There exist locally finite open covers (U j ) j∈Z >0 , (V j ) j∈Z >0 , (W j ) j∈Z >0 of R such that cl(U j ) ⊆ V j , cl(V j ) ⊆ W j , and cl(U j ), cl(V j ), cl(W j ) are compact, for each j > 0, where (W j , R n j , ϕ j ) is a local chart of R induced by the Hilbert map (2).
Proof.From Lemma 5, we know that there exist countable open sets  for (t, y) ∈ I × R. It is obvious that F defines a homotopy from h to f .Hence, the result follows immediately.

Conclusions
In this paper, we have considered the problem of approximating continuous functions by smooth functions on a subclass of singular spaces-subcartesian spaces.We have investigated a special class of subcartesian spaces-the orbit space of the proper action of a Lie group on a smooth manifold.By taking advantage of the geometric structure of the symmetry of the smooth manifold, we have shown that continuous functions on the orbit space can be approximated by smooth functions.In the future, we would like to investigate more subclasses of subcartesian spaces on which the approximation theorem holds.
Funding: This research was funded by NSFC (grant number: 61703211).